These terms from biogeography are becoming relevant to us because of the present-day rise of microbiogeography. More and more, we hear about the "geographic" distributions of microbial species and strains.
Sympatric speciation means that two or more species arose from a population living within the same region, whereas allopatric tells you that physical separation was involved, with the species arising in different locales. Both terms derive from the Greek patra, homeland (as in patriotic). Sym denotes same, alike, or similar; allo, other.
Zebras and quaggas (some sympatric). Source.
For animals and plants, "homelands" may be readily defined, as in remote islands, atolls, patches of forest, etc. To microbiologists, the "homelands" can be harder to define. They may be as far-flung as continents, or as close as different segments of the human intestine. However the region is defined, the high rate of mutation and lateral gene transfer makes it easy for us to assume that a microbial species often arises sympatrically. All it takes is some variation in the selective factors operating within that ecological niche.
But these are oversimplifications and there is more to it. For a thorough and thoughtful explanation of these and related terms, see a timeless 2007 post, one of the Basic Concepts series in John Wilkins' splendid blog, Evolving Thoughts.