LogFAQs > #969494367

As implied, they give different support bonuses when within support range. Characters get both their own and their partners affinity bonuses.

Supports will give characters some combination of Attack, Defense, Hit Rate, and Evade. Earth, for instance, is just straight Evade, 5% per rank (so 15% at max), so it's really good for faster frontliners like Ike and Oscar. Both Ike and Oscar are Earth affinity, so they both get +30% Evade from being within range of each other at A Rank.

Another example would be if you support Boyd (Fire) and Mist (Water). Fire gives 1.5 Attack and 7.5 Hit, while Water gives 1.5 Attack and 1.5 Defense. So Boyd and Mist would both get 3 Attack, 1 Defense, and 7% Hit (decimals are always rounded down), which makes Boyd hit harder and more often while letting him frontline a bit better because of his low Def, but does very little for Mist usually.

That said, you really don't need to care that much outside of higher difficulties or challenge style runs (ironmans, low turn count, etc)