These are means by which one species becomes multiple species. The difference is actually pretty simple:

In allopatric speciation, a population becomes separated by some kind of geographical barrier that prevents genetic mixing. A simple example would be a pond that was once part of a river. The animals on either side of the barrier would now be subject to selective pressure independently. This is the most common type of speciation.

In Sympatric speciation, species diverge without physical separation. One of the best studied examples of this is the rift lakes of Africa. Cichlids underwent an adaptive radiation where one species became 100s over a relatively short period of evolutionary time. Here, for example, individuals become segregated by feeding, with some preferring food present in the water column, while others prefer food buried in the sand. Over time, these groups became reproductively isolated and continued on a different evolutionary trajectory.