The terms both refer to cells that make up living things, and the main difference is in the definition of both--prokaryote means a cell without a nucleus, and eukaryote means a cell with a nucleus ("eu" means true.)
Prokaryotes are unicellular organisms such as bacteria. Their DNA is not contained within a nucleus, so duplication of DNA takes place in the cytoplasm. Eukaryotic cells make up multicellular plants and animals; there are also some that are unicellular organisms. Their DNA is contained within a separate area, the nucleus. DNA replication takes place there, and RNA leaves the nucleus, with protein synthesis taking place in the cytoplasm.
Prokaryotes are, in general, simpler than eukaryotes; eukaryotes have more separation of function.