The domain, (also called superregnum, superkingdom, empire, or regio), is the highest classification level in taxonomy. There are three domains in the modern taxonomic system. They are the Archea, the Bacteria and the Eukarya. The domain system was designed by Carl Woese, an American microbiologist and biophysicist. The first two are all prokaryotic microorganisms, or single-celled organisms whose cells have no nucleus. All life that has a nucleus and membrane-bound organelles, and most multi-cellular life, is included in the Eukaryota.
Archaea are prokaryotic cells which are typically characterized by membranes that are branched hydrocarbon chains attached to glycerol by ether linkages. The presence of these ether linkages in Archaea adds to their ability of withstanding extreme temperature and highly acidic conditions. Halophiles, organisms which thrive in highly salty environment, and hyperthermophiles, organisms which thrive in extremely hot environment, are examples of Archaea.
Even though bacteria are prokaryotic cells just like Archaea, their membranes are made of unbranched fatty acid chains attached to glycerol by ester linkages. Cyanobacteria and mycoplasmas are two examples of bacteria. They characteristically don't have ether containing linkages like Archaea, they are grouped into a different category - and hence a different domain. There is a great deal of diversity in this domain, and between that and horizontal gene transfer it is next to impossible to determine how many species of bacteria exist on the planet.
As the name suggests, the Eukaryotes are eukaryotic cells which have membranes that are similar to those of bacteria. Eukarya is further grouped into Kingdom Fungi (yeast, mold, etc.), Kingdom Plantae (flowering plants, ferns, etc.) and Kingdom Animalia (insects, vertebrates, etc.) and the now-defunct, paraphyletic Kingdom Protista (algae, protozoans, etc.). Not all Eukaryotes have a cell wall, and even for those which do, the walls don't contain peptidoglycan, which bacteria do have. While cells are organized into tissues in case of kingdom Plantae as well as kingdom Animalia, the cell walls are only found in the kingdom Plantae.