Fc is a community you join with certain boni that can be activated (like increased fighting XP or crafting XP reduced gear wear,....), ranks that give you rights for housing/invitations to the FC and activating those boni. Besides that you can set a personal message that only FC mates can see. You can only be in 1 FC at a time.

Linkshells are pure chat channels the "Master" can assign also here certain persons a "leader" so they can invite and kick but thats it. You can have up to 8 Linkshells.

Common is to be in a FC with your closest friend or raidgroup and have the linkshells for all other purposes like helping each other, chatting cross FCish, Hunting, trading,....