The Rubik's Cube can exist in 43,252,003,274,489,856,000 different permutations ("but only one solution!"). This is one way to arrive at that number:

There are eight corner pieces, each with three sides. The first corner (the order is arbitrary; pick any corner to be "first") can be rotated into three positions (since it has three colored faces) and can therefore exist in 8 x 3 = 24 possible positions. The second corner can exist in 7 x 3 = 21 possible positions, and so forth. However, it is impossible to rotate one corner without rotating any of the others, so the eighth corner can exist in only one position. Taking just the corners, we have:
24 x 21 x 18 x 15 x 12 x 9 x 6 x 1
= 88,179,840 possible positions
However, it is also impossible to switch two corners without switching two edge pieces (although it is possible to switch two edges without altering any corners). This reduces the number of possible positions by half, to 44,089,920.

There are twelve edge pieces, each with two sides. The first edge can therefore exist in 12 x 2 = 24 possible positions, the second in 11 x 2 = 22 possible positions, and so forth. However, it is impossible to flip just one edge without flipping any of the others, so the twelfth edge can exist in only one position. Taking just the edges, we have:
24 x 22 x 20 x 18 x 16 x 14 x 12 x 10 x 8 x 6 x 4 x 1
= 980,995,276,800 possible positions

There are six center pieces, but these cannot be moved in relation to each other. Mathematically, it doesn't matter which one is on top, so there is only one possible way to arrange the centers.

The final number of possible combinations is:
44,089,920 x 980,995,276,800 x 1
= 43,252,003,274,489,856,000 possible positions
...always assuming, of course, that you don't take any of the pieces out. (It's not too hard to find information on how to solve a Rubik's Cube, anyhow.)