Standard Sudoku: This is the original sudoku game (also called Number Place). It is also referred to as a 3×3 (3 mini-grids across, 3 mini-grids down). Each mini-grid, row, and column must contain the numbers 1 through 9. There can be no duplicates in any row, column, or min-grid.
Sudoku 4×4 and 5×5: This is similar to a standard sudoku, but there are either 4 mini-grids across and down, or 5 mini-grids across and down. The 4×4 variation usually uses the numbers 1 through 16, though some versions also add in letters. The 5×5 version uses both numbers and letter. Again, no duplicate letters or numbers are permitted in rows, columns, or mini-grids.
Sudoku-X: The X factor in this puzzle is simply the addition of one rule: Each of the two corner-to-corner diagonals must not have duplicate numbers. So, in a 3×3 sudoku-x, each column, row, mini-grid, AND the two diagonals will have the numbers 1 through 9.
Alphadoku: Similar to a standard sudoku but uses letters of the alphabet instead of numbers. How many numbers used will depend on how many mini-grids are across and down.
Samurai Sudoku: Typically, 5 standard sudoku puzzles are joined together in the middle. The first 4 full sudoku puzzles are placed separately, with the fifth puzzle placed in the very center sharing a mini-grid with each of the other 4. Each full puzzle can be worked separately, but they must all follow the rules of the standard sudoku.
Killer Sudoku: A killer sudoku requires simple adding. There will still be rows, columns, and mini-grids, but the individual squares (or cells) of the puzzle will be connected (either by color or by dotted lines). There is a small number in the upper corner of the connected cells. The numbers that go in each of the connected cells must add up to this number. Killer sudokus still follow all the same rules of unique numbers.
Irregular Sudoku: These puzzles do not have square mini-grids; they are "irregular" in shape but must still contain all the numbers 1 though 9. Standard row and column rules apply. While these puzzles are not any more difficult than a standard sudoku, it does take a while to "train" your eye to recognize an irregular mini-grid shape.
As time goes on, sudoku authors create new sudoku variations and even combine the above sudoku games into a new puzzle. Imagine a Killer Sudoku X Samurai with Irregular mini-grids. Now THAT sounds like a challenge!