There are four recognized domains of life, and eukaryotes belong specifically to one of these domains called Eukarya. The three other domains are Bacteria, Archaea, and a set of well-defined organisms classified under Eukarya.
Eukarya is characterized by organisms that have complex cells with a nucleus and other membrane-bound organelles. This domain includes a vast variety of life forms, such as animals, plants, fungi, and protists. So while there is one eukaryotic domain, it is part of a broader classification of life that includes three other domains, making the total four.