Eukaryotes make up one of the three domains of life, the other two being Bacteria and Archaea. Whereas the latter two domains are restricted to unicellular organisms, eukaryotes account for all multicellular life on Earth (in addition to some unicellular organisms as well).
See the figure below for a cross-section of a typical eukaryotic cell. The defining characteristic of eukaryotes is that their cells contain complex organelles encased in membranes, in particular the cellular nucleus. These organelles facilitate a division of labor of sorts, which is what enables the cell to carry out relatively complex tasks and has facilitated eukaryotes' miraculous leap from unicellular species to more complex multicellular organisms.