...and I thought you've already played FE...
*sigh*
Support gives you bonuses whenever two characters stand next to each other.
First of all, you need to make two certain characters stand next to each other for a certain number of turns (some characters need about 80-100 turns) When they stood there for that number of times, you get the "support" action when you press on the character. Read the dialogues between the two characters (sometimes they reveal something from the past or something about the future). When it's finished, you get a support level. C is the lowest and A is the highest. You can support with the same character for 3 times only. You can only have 5 supports at a time. That means you can only have 5 C supports, 3 A supports and 2 B supports, 1 A support and 2 C supports, 2 B supports and 1 C support, etc. Support provide your characters bonus depending on their affinities.
An A support with Fire x Fire saffinity gives you +15% in Strenth, +15% in Critical, and +15% in Evade
There are many other combinations...
Supports can also change the ending between the two characters