Domain is the largest classification category consisting of Archaea, Prokarya and Eukarya.It is a more recent category. All bacteria were once lumped together into Monera however, now there is a distinction between Archaea and Prokarya. More extremophile and primitive bacteria like organisms are placed in Archaea, while the other bacteria are placed into Prokarya. The older taxonomic categories included Kingdoms consisting of Animalia, Plantae, Fungi, Protista and Monera.These can be subdivided into Phyla. For instance, within the Animal Kingdom, if the organism has a notochord, it is placed in Chordata, if it is an invertebrate, it can be placed in Porifera(sponges), Cnidaria(jellyfish), Nematoda(roundworms), among several other Phyla. A phylum is a subdivision of a kingdom. A phylum can be divided into classes, these can be divided into orders, then families, genus and species. The genus and species is the two-name(binomial nomenclature) designation unique to an individual organism. To be classified into a species, means one can interbreed with other members of that species successfully to produce fertile young. In other words, all members of a species are the exact same organism. For example, Felis leo (lion) and Felis tigris(tiger) are both in the cat genus but their species are different.
The first 4:
Domain, Kingdom, Phylum and Class.