Archaea are very primitive prokaryotic cells that live in extreme habitats such as hot springs, extremely salty areas, etc. However, they have a much broader range of habitats than previously believed and play a role in the carbon and nitrogen cycles. They have no nucleus or membrane bound organelles. They reproduce asexually, but do not produce spores as seen in eubacteria. Eubacteria or "true bacteria" evolved independently from Archaea from an ancient common ancestor. They are involved in processes such as decompositon, the nitrogen cycle, make up a huge amount of biomass on Earth, allow for fermentation of foods like cheese and wine, cause diseases and live in and on other organisms. They have no nucleus and very rarely have membrane bound organelles.
Right they are both prokaryotes. The term bacteria was traditionally applied to all microscopic, single-celled prokaryotes.
However, molecular systematics showed prokaryotic life to consist of two separate domains, originally called eubacteria and archaea that evolved independently from an ancient common ancestor.