Oh bishop is not that hard! Still can't get anything for rook though ..
What can every other piece do that a pawn/knight/etc can't do?
Queen can't move to cause discovered check
King can't be taken/move into check
Bishop cannot leave its own colour
Pawn cannot move backwards
Knight can't give discovered attack
Rook cannot control more than 14 squares (besides the one it is standing on) at once.

Knight can't move to a square that is the same color as its current square.
In the starting position a rook can't make it to e1 (or e8 for black) in 5 moves or less?
Rook seems hard, not sure.

Queen can't move to cause discovered check
King can't be taken
Bishop cannot leave its own color
pawn cannot move backwards
knights must change color every move
Rooks cannot change rank and file at the same time
Eldridgium was pretty close, but kings and pawns cannot give discovered attacks, and pawns, knights, and kings cannot control more than 14 squares at once also ;)

For rook, how about jump over another piece ie. King during castling and land on the same colour square as it started (only applies to kingside castling). Sure the knight can jump over pieces too but it lands on a different colour square from where it started.
King cannot block check.
Queen cannot reveal a discovered attack.
Rook cannot change rank and file simultaneously.
Knight cannot move 2 squares diagonally in under 4 moves (like from a1 to c3--all other pieces can do this in 2 moves or fewer)
Bishop cannot change color of its square.
Pawn cannot have the option to move to more than 4 squares at a time.
I tried to choose less obvious ones
Also, the king can't be on the board twice. Or be pinned. And the bishop cannot do a real fork without capturing a piece. And knights cannot move to a square sharing a border with the square they are currently on. The queen can't have fewer than 21 legal moves on an otherwise empty chessboard. Pawns can't look differnent when rotated (if you're using a Staunton set). Rooks can't get through a connected wall of 8 pawns--in any shape as long as each pawn is within 1 rank of the pawn to the left and right of it--without capturing one.
Here's something interesting - The knight is the only piece that can be trapped by a king alone ( other than pawn ;) )

Trophy to jposthuma! Rook stumped me, but I could get the others. I had these for knight and king though:
Knight: Can't walk straight.
King: Can't give check.
btw. 9th eagle you are wrong i think. Here bishop did not capture a piece for a fork. ;)
Oh, that's true. Those types of forks rarely happen though. Well, the bishop is the only piece that if added to a K+P vs. K endgame can't necessarily force a win. (if the pawn is on a different square than the queening square).
Anyone wanna try? Answers needed for all six pieces. I know answers for pawn, knight, king and queen. But I can't figure out rook and bishop.
EG. What can every other piece do that the pawn can't do? ANSWER. Move backwards.

First correct answer for all six pieces gets a handsome "Have some tofu" trophy, which you probably haven't received before.