Introduced as one of the unique features of Beyond Earth, the "Affinity System" is actually somehow similar to Civ V's Ideologies and have nothing to do with the Social Engineering choices in Alpha Centauri. In SMAC terms, think of it as the path you have to follow to reach one of the victory conditions.
The guiding 'philosophies' behind Affinities will define the factions, the diplomatic relations between them, and will guide you to one of the available victory conditions.
Graphics wise, affinity choice will also affect your leader appearence as well as the looks of your cities and units.
Your faction always starts off human, but the decisions you make in quests and the technologies you choose to research will take your faction to the next stage of humanity. There are three very different Affinities –
Purity, Supremacy and Harmony - that represent some post-human ideas that the designers found to be the most evocative for them.
Purity – A faction that aligns itself with the Purity Affinity is a faction that refuses to be changed by the alien world. Instead, the faction is out to create the ideal human and change the planet to be more like Earth. Of course, the only way to do this is through brute force, dramatic terraforming, and big guns. The Purity units beautifully illustrate this faction’s unwillingness to change. The units are big and powerful, yet still have a classic, valiant look to them. They’re both extremely intimidating and majestic at the same time.
Supremacy – Going the Supremacy route gets you some of the sleekest and coolest units in the game. This Affinity is focused on developing advanced technology and implanting its people with cybernetics. The attitude here is that humanity has put a great deal of value into technology, because it’s what saved them and brought them to the new planet. Factions willing to go this direction gradually give up more and more of their organic forms, but this is by no means the ‘evil robot’ route.
Harmony – Factions who choose the Harmony Affinity have accepted the new alien world and embrace it. These players will not change the planet, but they allow the planet to change them right down to their very own DNA. Harmony units takes on a more organic look as the synergy between the faction and the alien life progresses. Devoting your research to Harmony will eventually allow you to get alien creatures on your team and breed new units based on their genes. You will team up with the planet, who at one point was bent on killing you, and go on the offensive against all your opponents.
As you accumulate points in the Affinity of your choice, you eventually earn Affinity 'levels' which unlock bonusses for your faction and upgrades for your units. Affinities are not mutually exclusive, as you can gain levels in all three of them, so most of the time, you'll have a 'dominant affinity' and two 'minority affinities'. Also, once you gain level benefits from the minority affinities, you will unlock 'hybrid upgrades' for your affinity-unique units.
Bellow is the list of the benefits and the affinity level needed to unlock them:
Purity:
Level 1: Aliens will not attack tiles with Explorer Units
Level 3: +20% Strength and Ranged Strength when attacking or defending against Aliens
Level 6: +2 Orbital Coverage in all Cities
Level 8: 4 Floatstone resources for free
Level 11: Dirty Bomb Operation can be performed by Covert Agents
Level 13: Exodus Gate can be constructed for Promised Land Victory Quest
Harmony:
Level 1: Alien aggression level returns to neutral twice as quickly
Level 3: Units take 5 HP less damage from Miasma
Level 6: Units heal 5 HP in Miasma
Level 8: 4 Xenomass resources for free
Level 11: Call Worm Strike Operation can be performed by Covert Agents
Level 13: Mind Flower can be constructed for Transcendence Victory Quest
Supremacy:
Level 1: Explorer Units can build an additional Expedition
Level 3: Roads and Magrails do not cost any Energy to maintain
Level 6: Orbital Coverage on and next to every tile with Firaxite
Level 8: 4 Firaxite resources for free
Level 11: Sabotage Operation can be performed by Covert Agents
Level 13: Emancipation Gate can be constructed for Emancipation Victory Quest