Affinity bonuses are gained when a pilot has completed 5, 10, and 15 missions in the same 'mech. Affinity will begin to decay after three missions spent in a different 'mech so it is possible to juggle two or sometimes even three on the same pilot if you can keep track of things.
At 5 missions the pilot gains the "Comfortable" bonus, giving them a +1 to their Piloting stat.
At 10 missions the pilot gains the "Attuned" bonus, giving them a +1 to all skills (Gunnery/Piloting/Guts/Tactics).
At 15 you unlock the good stuff: Special effects that vary depending on the 'mech your pilot is attuned to.
The following is the list of the tier 3 affinities and their effects as found in BTA. Most of them mimic one of the quirks which 'mechs come with. Unlike quirks, which we can think of as innate bonuses, affinities can be seen as the realised potential of committing to a specific 'mech. The effectiveness of tier 3 affinities can also be dependent on the baseline capabilities of the 'mechs; some affinities significantly change the effectiveness of certain 'mechs, while others do very little and can even be a chore to get, as the mechs themselves are pretty bad. They have the ability to save a 'mech from being total rubbish, to complement their strengths, or to make them fantastic.
Pilot Affinity: Agile
Gains 1 bonus evasion from movement
Pilot Affinity: Brawler
Kicks, punches, and physical weapon attacks deal 15% more damage
Pilot Affinity: Command Unit
Gains +1 initiative
Pilot Affinity: Cool Running
Adds 6 bonus heat sinking, effectively two standard heat sinks
Pilot Affinity: Encouraging
+2 morale whenever morale is gained
Pilot Affinity: Evasive
+2 bonus against being hit
Pilot Affinity: Ice Cold
Adds 9 bonus heat sinking
Pilot Affinity: Keen Eyed
15% boost to sensor and sight range
Pilot Affinity: Nimble
15% more jumping distance
Pilot Affinity: Precise Shot
+1 bonus on weapon attacks
Pilot Affinity: Resistant
50% less chance to suffer critical hits
Pilot Affinity: Stabilized
Reduces stability taken by 50%
Pilot Affinity: Sturdy
Gains 2 extra pilot hits
Pilot Affinity: Unlimited Power
Pilot Affinity: Zippy
15% boost to sprint speed!