What is the difference between allopatric and sympatric speciation?

Allopatric speciation occurs when populations of a species are separated by a physical barrier - this could be a river for animals that cannot swim, for example. Separation of the populations means that there is very low or no gene flow between them - the proportion of different genotypes in each population is therefore able to change independently of the other (there's no mixing up of genes between the two populations). Over time, these changes may be so drastic that the populations become unable or unwilling to breed with each other, and could therefore be described as a pair of species.

Sympatric speciation occurs without a physical barrier to gene flow. This is more common in plant species - plants can mutate in a way which results in them producing offspring with double or even quadruple the number of chromosomes they normally do. The sex cells (sperm and eggs) produced by these individuals cannot fuse with sex cells from a "normal" plant - the plants with unusually high numbers of chromosomes therefore become isolated gentically from the "normal" plants, even though they may be growing right next to each other. This genetic isolation results in the two types of plants developing into species due to lack of gene flow and independent changes in the genotypes of plant populations.

(A way to remember these - Allo- could sound like "Alps"; think about a mountain dividing two populations. Sym- sounds like "same"; think about species developing in the same place!)