The domain that contains organisms with a nucleus in their cells is called Eukarya.

Here's why:

* Eukaryotes: Organisms in the domain Eukarya have cells with a true nucleus, which houses their genetic material (DNA). They also have other membrane-bound organelles like mitochondria and Golgi apparatus.

* Prokaryotes: Organisms in the domains Bacteria and Archaea are called prokaryotes. They lack a nucleus and their genetic material is located in a region called the nucleoid.

So, the presence or absence of a nucleus is a key characteristic that distinguishes eukaryotes from prokaryotes.