These pages discuss some variations of the Rubik's Cube puzzle. Starting with a traditional 3x3x3 cube is probably best, and there are ample resources for it elsewhere, so I won't repeat any of that material. After you are comfortable with the 3x3x3 cube, you can explore the other variations listed here.
Notation
The cube has six faces, which will be called Up, Down,Front, Back, Right, and Left. Turning a face 90 degrees clockwise is represented by the first letter of the face's name, in upper case. For example, "R" means to turn the right face 90 degrees clockwise (such that the front right pieces become the top right pieces). A letter followed by a tick mark means to turn the face counter clockwise by 90 degrees (e.g. R'). A letter followed by the number 2 means to turn it twice (180 degrees).
Odd sized cubes (3x3x3, 5x5x5, 7x7x7) have a middle layer, but since the 6 center pieces never move with respect to one another, we can consider their orientation fixed and never spin the center layers (and thus we don't need any notation for them). The larger cubes (4x4x4 and above) have additional inner layers. These layers are notated by lowercase letters for the face they are closest to. For example "r" means to turn the layer just inside of the right face.
Parentheses are sometimes uses to group moves together for easier readability. For example: r (U' R U) r' (U' R' U).
Corner pieces have three colors are referred to by their three face names, for example UFR is the top-right corner on the front face. Edge pieces have two colors and are referred to by their two faces and optionally an inner layer. Proceeding from top-left to top-right on the front face of a 5x5x5 cube we have: UFL, UFl, UF, UFr, and UFR.
Center pieces have a single color and are identified by a single face and one or two inner layers. Thus the nine center pieces on the front face of a 5x5x5 cube are labeled:
Conjugation
I prefer to use a small number of simple general purpose operators as opposed to memorizing a large variety of specialized ones. One technique for making general operators even more useful is conjugation.
First, it is useful to define the inverse of an operation. Simply put, and inverse just reverses the operation. The inverse of a single twist is twisting the same face (or layer) in the opposite direction. For example, R' is the inverse of R and u is the inverse of u'. To invert a sequence of twists, you must go through the steps in reverse order and use the opposite direction for each twist. For example, the inverse of U R is R' U'. Note that 180 degree twists are their own inverses (the inverse of R2 is R2). If you perform a sequence of twists - no matter how complex - and then perform the inverse of that sequence, the cube will be restored to the exact state it was in before you started.
Let's say you have a useful sequence called Y that flips the UF and UB edge pairs without altering anything else on the cube. What if you need to flip some other two edge pairs? As long as you don't care about scrambling the rest of the cube, it is proably pretty easy to move the desired edge pairs to UF and UB. Let's call this sequence of moves X. Now consider what would happen if you performed X Y X' (where X' is the inverse of X). First you move the desired edges to UF and UB, but mess up the rest of the cube. Then you flip UF and UB while leaving the messed up portions perfectly intact. Finally you apply X' which moves the desired edge pairs back to their original location and also unscrambles the rest of the cube. Creating sequences of the form XYX' is called conjugation and is key technique for solving any Rubik's cube.
The Cubes