Cells in all three domains are based on the same basic foundations of cell structure, but there are key differences between the domains.
In all three domains, the hereditary material is DNA; their cellular metabolism is based on proton gradients which drive ATP synthesis (using the same protein system, ATP synthase); they all have phospholipid-based membranes, and they use protein catalysts (enzymes) to speed up metabolic processes. They use RNA and ribosomes for protein synthesis.
However, there are key differences between species in the domain Bacteria and the domain Archaea. These two domains have only prokaryotic species (unicellular organisms without a nucleus or other membrane-bounded organelles), so superficially they are very similar. But in certain molecular structures, there are more similarities between species in the domain Eukarya (the eukaryotic organisms) and Archaea than between Bacteria and Archaea.
Currently accepted ideas of the origin of eukaryotic cells are based on the model that an archaeal cell and a bacterial cell fused into a single cell, which then evolved into all eukaryotic species.