Eukaryotic cells (Domain Eucaryota, which includes multicellular life, like plants, animals, and fungi) are, on average, much larger and more complex than the earlier evolved prokaryote cells (Domains Bacteria and Archaea*). They have organelles, including mitochondria that power them and chloroplasts (at least among plants) that carry out photosynthesis. Their DNA is stored in a nucleus, and consists not just of genes (as in prokaryotes), but of large stretches of non-coding DNA (most of their genome), separating pieces of genes. The ancestor of present-day eukaryotes reproduced sexually, although some eukaryotes have since given up sex.
There are different ways that life increases in complexity. The origin of the Eucarya has something in common with a much later event, the origin of agriculture (check out September 10 on logarithmichistory). Starting 10,000 years ago, we Homo sapiens brought other animals and plants under our control, managing their reproduction, and selecting them (first unintentionally, then intentionally) to suit our purposes, until now most domesticated creatures couldn’t survive in the wild. Our own numbers and social scale increased enormously with the rise of agriculture.
At least 2 billion years ago, an archaeon cell gobbled up one or more bacterial cells (or was parasitized by them). The bacteria ended up surviving inside it, and after many generations became a kind of domesticate inside their host. Eukaryotes do domestication one better than humans: they carry their livestock inside their bodies. Eventually this domesticate evolved into mitochondria, the little power packs that pump out ATP for the rest of the cell to use as as an energy source. Over the course of time all but a small fraction of the original bacterial genome was moved into the nucleus.
In the last few years we have come closer to understanding how this momentous step occurred: we have discovered a new branch of the Archaean tree, the Asgard archaea. The Asgard archaeans carry some genes otherwise found only in eukaryotes, and it looks likely the first eukaryote to start hosting the bacterial ancestors of mitochondria was an Asgarder, or close relative. Just this year we finally succeeded in cultivating these creatures in the lab. (It was hard to do.) They are tiny little spheres with long filaments protruding from them. The partnership between Archaeans and bacteria may have begun with bacteria nestling in these protrusions.
Humans developed agriculture multiple times independently around the world. As far as we know, eukaryotes evolved only once, long after the origin of simpler forms. The evolution of eukaryotes might be a very unlikely chance event. The universe may be full of bacteria, but harbor more complex cells only sparsely.
* Domain Archaea, a billions-of-years-old group of single-celled organisms looking like bacteria but biochemically different, should not be confused with the Archaean Eon, a billions-of-years-long stretch of Earth history.