For most factions in the game, colonization in Endless Space 2 takes the established 4X game trope of sending a civilian settler (in this case, a colony ship) to a location, and then triggering a special "colonize" action that consumes the colony ship, and tries to turn a planet into an official part of your empire. There are a few factions that use slight (or extreme) twists on this mechanic, and those are listed below.
A player needs to mind overcolonization disapproval 
when expanding their empire, which incurs sharp penalties to happiness when establishing more colonies than a set amount. This is covered elsewhere, but depends on galaxy size, difficulty level, and faction choice.
Sending a colony ship[]
Every faction that utilizes colony ships starts with one immediately. Beyond that point, they needed to be built. A colony ship is its special civilian ship type, with no other being capable of utilizing the module. These ships have very limited attack and defense capabilities and should be escorted to their destination if there's any risk of attack by pirates or a hostile faction, which is quite low early game. It is recommended to upgrade the ship blueprint with better engines once the player have researched them.
Once the colony ship has reached its destination, it can establish an outpost on any of the habitable planets.
Establishing an outpost[]
Once the outpost has been established, it needs to be fed a certain amount of 
(or 
for Riftborn) before it becomes a proper colony.
The outpost itself will produce 5 and 10 whilst using up -10 
. (Note that if the played faction has the Continnum Sculptors affinity (default for Riftborn), the outpost will not produce food.)
Several players can establish outposts at the same time (which makes an action to leech some of the contestants' available) with the first player to reach the target amount winning control of the system. Upon winning this, all planets with outposts (including those of opponents) will be considered colonized by the winner, so it might be desirable, if it's clear that you're going to lose a race for a colony, to deliberately 'decolonize' to deny the opponent the benefit of trying to colonize the extra planet you were going for. Note that if you do 'decolonize', you will receive a free replacement colony ship at that system, albeit with only one engine equipped.
It is possible-but-rare for a tie to actually form, at which point the outposts in a system will be in a stalemate until one faction breaks the tie by weakening an opponent's colony (see notes below).
Once an outpost becomes a colony, it is possible to start building system improvements and colonize other planets in the system.
Growing an outpost into a Colony[]
To transform into a colony, the outpost needs to accumulate enough to grow a single pop unit (300 on normal speed). For Riftborn, they would instead need to accumulate (300 on normal speed).
This comes from three sources: The base food production that the outpost has (5 ), and the natural production value that a single generic pop unit (and any raw output bonuses) would give from a planet, and from periodically-departing small (visible in-game) civilian supply ships which will depart from a nearby colonized system you own.
For Riftborn, the comes from two sources: The base industry production that the outpost has (10 ), and the natural production value that a single generic pop unit (and any raw output bonuses) would give from a planet.
Natural Production[]
Non-Riftborn factions[]
The growing outpost is considered to have a single population unit of a "generic" species, with no traits, whose only purpose is to grow (although it does also exploit and 
). The growing outpost has no System Improvements (However, the outpost itself will produce 5 ), but anything else that boost production will affect this generic pop - this includes planetary Anomalies and Resources, Laws, Empire Upgrades, Assimilation Bonuses from Minors Factions, and even Behemoths with FIDSI-boosting modules in orbit. It is also, thus, usually ideal to place your outpost on the most fertile world possible, ideally an actually Fertile 
world.
So for example - if the planet has any anomalies, the Ecologist "Green Fertility Bill" will boost it by +2 (3 in republic), and regardless of anomalies the Ecologist "Cool Copies" clause will boost it by +2 (+4 in republic), which sometimes can end up doubling the natural output of many planets if both laws are in force. The Scientist party later on gets a +30% law; this affects colonies. There are also enormous bonuses possible from minor faction assimilation effects; Sowers can add a +25 on a sterile world, Basryxo will add 10% of the value into the value, Haroshems can guarantee a minimum of 5, etc.
The second boost option on the outpost (the one that costs ) will double the total output of this process, after all other factors have been taken into account. The cost scales linearly with the number of systems you already own. The formula cost being (50 + (100 x amount of systems) ).
This implication also means that any Luxury Resources or Strategic Resources can be exploited with their deposit value (as long as the technology required for exploitation has been researched for the latter). Some Laws can even increase the production beyond their base deposit value (notably, the Tier 3 
law for Luxuries and Tier 2 
law for Strategics).
For Riftborn faction[]
The Riftborn follows largely the same principle, except that instead of utilizing the value and ignoring the value, they have it the exact opposite, utilizing the value while ignoring the value.
The outpost itself produces 10 , while adding any value they have from the planet too. That typically means that if fast colonization is the goal, colonizing Hot 
planets is the way to go. (More information with regards to Riftborn is covered down-below in "Other Approaches" section)
Supply Ships[]
An already-colonized system (typically your highest -producing system) will be designated as the supply system for the colony, and will send out a fraction of their production in supply ships every 2 turns to help grow the colony.
While it's supplying the outpost, the supply system will lose 40% of its production: 20% is loaded onto a supply ship, while the other 20% is wasted (presumably in transit). Every 2 turns, the supply ship will take off from the supply system and make its way to the outpost. The supply ship is a civilian ship that is visible in-game, has a +1 
bonus (like the 
Colonizer ship) and has it's movement speed affected by all modifiers applicable (such as the +2 for Sophons, and the +3 (+4 if Republic) bonus from Tier 2 
Law, and the movement speed bonus granted by unlocking Level 3, 4 and 5 of Empire Development which grants +1 to "Civilian Ships" for each level unlocked.).
This ship can be stopped by enemy blockades and then be destroyed by enemy fire (yielding a bounty equal to the value carried by the ship). A maximum of 3 supply ships are sent out at a time.
You can re-assign which system is the supply system at any time. This is desirable because first, it's often imperative that the colony ships reach the target planet in a reasonable amount of time (or avoid enemy territory), and second, the food drain from the supply ships can be significant, and can tip a system into negative production. To avoid starvation, make sure the supply system is happy/ecstatic, and/or assign a Hero who boosts production.
The third boost option on the outpost (the one that costs 
) will double the taken from the supply system and delivered to the colony. As this means the supply system is losing 80% of its production, this can easily tip the system into starvation and cause it to lose several population over the 6 turns of supplying, so be careful using it. Assuming the supply system is reasonably close, however, it can be very effective at quickly growing a food-poor outpost.
Contesting Colonies[]
Multiple factions can colonize different planets in a system. Each outpost will grow independently without any interference from the others. The first boost option on the outpost (the one that costs 
) causes your outpost to steal 10 /
from all opponent's outposts. This can tilt a weak enemy outpost into negative growth. They can, however, do it right back to you, which just evens out the trade. (Note that if Riftborn do this, the stolen growth is added to their colony as growth, and vice-versa.)
If you can blockade a system with military ships in orbit, they will suppress the growth of the enemy outpost, reducing the outpost's food growth by the ships' siege strength (the amount of 
they'd destroy per turn). Blockading a system also prevents all "boosting" actions, including that spending of military manpower mentioned above, and prevents inbound supply ships from being able to reach the surface. (They will orbit the system until the blockade ends; you can choose to attack them for the bounty.) Note that blockading an outpost is considered a moderately hostile action to the AI, and will immediately give them a fairly negative stance towards you - this will wear off after a while, but can easily tilt the scales.
The first outpost to fill their supply will, after stalling for a single turn (to allow ties), win the race and grant the system to the winner as the full "owner", giving them not only the planet they colonized, but any other remaining outposts.
An outpost whose stockpile of is driven below zero will fail and be removed. This often is a deliberate side-effect of blockading an enemy colony.
Note also: any situation where multiple contestants were vying for colonization will leave "partial" pop units of their type on the system, which will be picked up and grown to a full unit in the regular pop growth sequence. This means that if Sophons and Lumeris both had outposts on a system, and the Sophons won the contest without eliminating the Lumeris outpost, the Sophons will subsequently end up growing a Lumeris pop as their second or third pop unit, and from then on can grow Lumeris pops normally. This is particularly relevant if contesting colonies with Cravers, as you will eventually receive a Craver pop on the system (that will populate and accelerate the depletion of your planet(s)) unless you completely eliminate their outpost.
Winning outpost races against an AI will "annoy" them; it's considered a very minor act of hostility. Doing it once will be forgotten, but repeatedly doing this will make them belligerent, at least if you're in a cold war diplomatic stance (as you are at the start of the game).
Colonizing planets in a system[]
Once the system itself has been colonized, the player can colonize other planets by adding the project to the building queue. After the project is complete, it's possible to exploit the planet and move the population from other planets in the same star system to it and vice versa.
Colonizing most types of planets requires researching specific technologies - typically, the less hospitable, the more advanced. Riftborn consider planets to have opposite hospitability from others.
If you manage to have a planet colonized without the specific tech for that planet, you will suffer a -50% 
/
output on that planet until you have the technology researched. This penalty can be negated to -25% / by having 
Ecologists as the main political party (and therefore, their Power Law "Hardship Ready") and a Government of Dictatorship/Republic. Nakalim's special Planetary Specialization "Temple to the Lost" negates the penalty completely.
Other approaches[]
All of these faction-unique mechanics are described in far more detail on the specific pages covering the faction, but they're briefly described below:
Factions with Standard Colonization Ships, but Different Colonization Mechanics:[]
Riftborn[]
As the Riftborn are incapable of producing , their outposts are instead grown via the value on the target planet, but otherwise function the same way (they need to reach 300, and can be siphoned by other factions via troops, and/or can siphon from others - yes, they're siphoning the value of another faction and treating it as a boost to their value).
This can be boosted by paying strategic resources (either 
or 
), which respectively will either boost the planet's value by 100%, or boost the value by +25. These mutually stack with each other (the 100% doubles the +25).
Unlike other factions, the riftborn do not send out colony-supply ships, nor do they lose on a supply world. As with other factions, since their production towards a colony is based on the actual values on the target planet, anything (like laws or assimilation bonuses) that boosts the production values on a planet will boost colonization speed, except for them, it's based on the value rather than . There are a few notables here - the standout one being the Pulsos "Thinkers and Tinkers I" which provides a potentially enormous gain, reducing all colonization to a few turns.
Hissho[]
Hissho produce colony ships and create outposts like other factions, but their outposts have to be manually turned into colonies by spending 
. This action can be taken as soon as the outpost is created, but its cost will decrease as the outpost grows, down to a minimum of 10 plus a penalty based on Overcolonisation.
Alternatively, Hissho may collect 
from unoccupied planets using mining probes. In this case, resources from planet will be added to pops in their main system.
Factions with Colonization Ship Substitutes:[]
Vodyani[]
Vodyani are ship-bound and do not colonize planets at all, instead living on 
Arks. Arks are colossal residential ships that essentially "house" everything another faction would typically build on the planet's surface - an Ark can dock/anchor with a system to enter resource production mode (when travelling, they produce nothing), but is otherwise free to travel wherever and whenever it wants. Every Vodyani on an Ark "projects" on every planet in a system it is anchored to, provided the "colonization" tech for the appropriate planet type has been researched.
In order to add a new system to their empire, the Vodyani simply build a new Ark in a system they already own, pilot it to the desired destination, and anchor it there.
Unfallen[]
Instead of colony ships, the Unfallen utilize 
vineships. A vineship functions similarly to a colony ship, but is reusable and has a much higher cost. Vineships can spread vines from nearby system that has already been vined, which takes a number of turns. Systems are considered vineable if they are connected to vined system via starlane, or are in range of 23 dots from it. Multiple vineships can place vines faster. The envining process can be paused for up to 3 turns, after which it is reset. Once the Unfallen player has spread vines to a system, they can pick a planet from that system to colonize for free, instantly.
As of the current patch - a total of 15 vineships are needed to reach the fastest vining speed of 1 turn.
Vaulters[]
The Vaulters do not build colony ships or create outposts. Instead they use their unique ship, the Argosy, to instantly turn a colonizable planet into a colony. This will also trigger a Golden Age on the system, granting FIDSI bonuses. After colonizing a system, the Argosy has to recharge for several turns, though this recharge period can be bypassed by spending and Strategic Resources. Vaulters cannot colonize on a system that has outposts on it.
Factions without Colony Ships:[]
Lumeris[]
The Lumeris do not build colony ships; they instantly purchase outposts with (Formula: 200 x Dust Inflation) Other than this they work identically to other factions - their only other unique mechanic is that they can sell outposts they create, via diplomacy. Note that if you're playing against a Lumeris AI, they are reasonably receptive to selling their outposts, as long as you're not hostile.
Umbral Choir[]
Umbral Choir faction has a very different system in terms of colonization. Instead of having colonies they have a unique system linked to sanctuaries that may be placed on any system (whether or not presence of an enemy faction). Sanctuaries are acquired through the new mechanic introduced from penumbra extension which is Hacking. After a system is hacked, one of the outcome is the option to place a sanctuary then it can be filled only with the home population of the Umbral Choir.