This is bughouse. I don’t play this particular version, however. You can check the king with a dropped piece (thus making knights unusually powerful in attack).
We used to play that in high school, but we called it “Super King”. Another variant: the player with the Super King also gets three pawns, which can also make two moves at once. If one advances to the last rank and becomes a Super Queen, the game is pretty well over.
We also used to play “Fox and Hounds”, which is played with one black pawn and four white pawns. They start on the same color square (say, all on white or all on black), on the nearest rank of the board for each side. The pawns only move diagonally, one square at a time. The “fox” (black pawn) has to evade capture. The “hounds” (white pawns) have to trap the black pawn by blocking him in, either against the side of the board or by blocking all four diagonal moves. No jumping. It’s really just a time-waster, something to do when you don’t have time to even set up all the pieces.
Edit: we apparently played “bug house”, but we called it “Siamese Chess”. One rule was, when a pawn becomes a queen, it’s simply laid on its side. If that queen-pawn is captured, it goes to the other board as simply a pawn. Also, there is no checkmate; the king can actually be captured, and the board where the king (or even both kings) is captured can continue to play and supply the other board with pieces. The game is only lost when both of one team’s kings are captured. To make this even more interesting, you’re not required to announce check.
Shatranj: pawns have no double move, there is no castling, Rooks are the only line-pieces. Instead of Bishops, fils, which leap exactly two squares diagonally (i.e. whatever is one square away on the diagonal does not obstruct it). Instead of Queens, firzans (move one square diagonally). Pawns promote only to firzans. As well as checkmate, a player can win by bare King - taking all the enemy’s men apart from the King. The fil is a miserably weak piece as it can visit only eight squares on the whole board. Strategically, it is important that the four fils cannot attack each other or defend against each other, and neither can the opposing firzans.
I’m assuming you mean variants on chess that don’t require special gear. In which case, I rather liked pole chess, which I learned to play in high school from a friend who was more of a geek than I am (both of us heard about it first in a book by Piers Anthony); he claimed to have found the ‘official’ rules, but I can’t find anything about it via Google. Regardless, it was pretty fun.
My friends and I played a variant we were told was called Kings Chess. By sacrificing a piece, you could grant your King that piece’s movement rules, though he still had to follow the check rules. It made for quick, brutal games. I particularly remember one clever move by an opponent, who positioned his king diagonally from mine (I had pieces blocking my king at the time) and sacrificed his bishop on the next turn. Insta-checkmate.
Then there was Traitor Chess. Once per game you could ‘turn’ one of your opponents pieces. Your own piece of that type died (bishop for bishop, for example), but you gained control of his, wherever it happened to be. Kings and Queens were immune.
If other stuff IS allowed, I had LOADS of fun with Knightmare Chess and Doubles Chess. The interesting twist to Doubles Chess is that you’re not playing against the person on the other side of the board; rather, you’re playing two games at once against the people on either side of you. The one across the board is your ally.
Maybe not exactly what the OP had in mind, but my favorite chess variant as a kid was Archon.