Meiosis
Before it begins meiosis, the cell replicates its DNA, so it has four copies of each chromosome; two identical copies of each chromosome, you inherited from your mother, and two identical copies of each chromosome, you inherited from your father. The two identical copies of each chromosome are called sister chromatids, and are actually physically attached to each other to form the classic X-shape you often see in drawings of chromosomes. The two sets of sister chromatids are called homologous chromosomes because, even though they form a pair, they are not identical.
Meiosis I
During meiosis I the parent cell forms two daughter cells. Each daughter cell inherits one of the two homologous pairs of each chromosome -- a pair of chromosome 1 sister chromatids, a pair of chromosome 2 sister chromatids, and so forth. The part of the process where the homologous pairs actually separate is called anaphase I. In anaphase I, filaments called microtubules pull the homologous pairs apart, so one pair of sister chromatids goes to one end of the dividing cell, while the other pair goes to the other.
Meiosis II
Meiosis II is similar, but now the sister chromatids are separated so each daughter cell ends up with one of the two sister chromatids from the pair it inherited. Again, the actual separation takes place during anaphase II, where microtubules pull the sister chromatids apart and toward the two opposite ends of the dividing cell. The net result of meiosis is a collection of four daughter cells, each with only half as many chromosomes as its parent. A daughter cell has only one copy of chromosome 1, one copy of chromosome 2, and so forth.
Recombination
One more key difference between the two stages is that, in meiosis 1, just prior to anaphase, pieces of each chromosome can be exchanged between the two homologous pairs. This process is called recombination. Ultimately, each of the four daughter cells produced by meiosis can end up with a copy of the chromosome you inherited from your mother, a copy of the one you inherited from your father, or a "recombinant" that includes pieces of both. Each of these three possibilities holds true for all 23 chromosomes, so a gigantic number of combinations are possible.