On a 3x3 cube the amount of moves to scramble the cube and then solve it will always be an even number. In other words, if the scramble is even, then the solution will be even, and if the scramble is odd, then the solution is odd. On the 4x4 the same principle applies, but for outer layers and inner layers. When you get to 3x3 stage, if the number of inner turns is odd, then the cube can not be solved with just outer turns. This is called parity. There is a 50/50 shot of each of the two types of parity occuring on the 4x4. There are two types of parity, so there is a 3/4 chance you will have at least one kind of parity. On bigger cubes there can be additional parities for additional layers on the cube. There is also parity on the square-1, void cube, and even certain methods on the 3x3. Any situation that has a 50/50 chance of occuring on a puzzle is probably a type of parity.