Here's a weird one: Babson Task. It's a directmate in which a Queen promotion by the defender must be met by a Queen promotion from the attacker, a Rook promotion must be met by a Rook promotion, and the same goes for Bishop and Knight. All of White's promotions are the only moves that win in the right number of moves, although it's beyond me why the heck it works or how the composer came up with it in the first place.
Terminology, and other interesting stuff.
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Yeah, famous puzzle.
It was thought to be impossible, until an unknown Russian soccer coach came up with it in the 80's. There are a few variations now.
And then there's the maximummer, a personal favourite.
Both sides have to make the geometrically longest move, I'm not gonna type in those moves again, here's the link
http://www.chess.com/forum/view/more-puzzles/maximummer
Double zugzwang is always fun :)
There are a lot of strange concepts, words, terms in composing.
If you come accross any, please feel free to post them here, with examples if you can find them.
First, some basics.
Direct mate, help mate, self mate.
Direct mate:
A normal chess position, White to play and mate Black in x number of moves.
Help mate:
A helpmate is a kind of chess problem in which both sides cooperate in order to achieve the goal of checkmating Black. In a helpmate in n moves, Black moves first, then White, each side moving n times, to culminate in White's nth move checkmating Black. In a helpmate in two (sometimes abbreviated to h#2), for example, the solution consists of a black move, a white move, a black move, and then another white move, giving checkmate. Although the two sides are cooperating, all moves must be legal according to the rules of chess. (wiki)
Example h#2
Self Mate:
White moves 1st, and forces Black to mate in x number of moves.
Example s#2