The notation used in these lessons is very widespread and accepted. Speedcubers will usually use an extended notation, which also includes movements of two layers at the same time or rotations of the whole cube.
The symbols in the standard notation are the first letters from the names of the sides. So we have
Colors, or the side of the cube you're looking at are actually not relevant for the effect of an algorithm. Now, 1 move, e.g. R is made as follows: first, look at the R(ight) side of the cube. Then, turn the face your looking at clockwise 1/4 (as in the animation above). As you can see, if you do that 4 times, the cube will be restored to its original state; yours is scrambled, right?
Additionally there are three other important symbols, that are used to describe rotations of the complete cube. They are named after the mathematical X, Y and Z axes:
x - rotate cube looking at Right face (also [r])
y - rotate cube looking at Up face (also [u])
z - rotate cube looking at Front face (also [f])