The honey bee is the common name for the genus Apis. The most commonly thought of is Apis Mellifera, the European Honey Bee, which is the species most often kept by humans. Currently they are all under attack by the varroa mite, originally hosted by Apis cerana that has nearly wiped out wild hives and hurt many beekeepers.
The European Honey Bee is also subdivided into races or sub-species. The most common is the Italian, with their distinctive bright yellow and black markings and large size. They produce fairly large amounts of honey, but are more aggressive than some other breeds. Also on the plus side they tend to be more resistant to most bee diseases than others.
Second most common is the Caucasian Honey Bee from the Caucusus Mountains. (Naturally.) They are darker in color, usually having grey and black markings. They are gentler than the Italian Bee though they also tend to glue internal parts of the hive together and so many beekeepers will not keep them.
The third most common race kept in North America is the Carniolan Honey Bee. Their home is generally the upper Danube River valley. They are large and also generally grey in coloration. One problem is they tend to be smaller producers of honey.
German or Black Bees are good producers of honey, but they are even more aggressive than Italian Bees and more susceptible to foulbroods. Because of this susceptibility almost no one in North America keeps them today. On the positive side they tend to use less propolis than other races.
African or Africanized Honey Bees are the notoriously aggressive breed native to various mountain ranges in Africa. Several queens were imported to Brazil in 1956 and they have spread from there to the southern United States and northern parts of Argentina. They cannot stand cold climates so it is not expected that they will spread much further north into America, though they may interbreed and make current strains more aggressive. They tend to attack in mass and this makes them much more dangerous than most other breeds.
Cape Bees are from South Africa. They are not as aggressive as some other sub-species. They are most noted for their ability to start laying eggs when the queen is killed. This makes them unique among honey bees and very studied. Otherwise they are not very notable.
There are other breeds, but most are not as notable as these are or as widely kept.