This common ancestor must have been very primitive, of course. One is tempted to think it might resemble modern-day religious fundamentalists, but in fact it was probably even more primitive, if you can imagine such a thing.
It's not absolutely clear there was actually one common ancestor, but that's what evidence currently indicates. But assuming there was, it's fascinating to speculate about what this ancestor was like.
Here's a very detailed blog post that discusses the issue: Ur... Again (Sort of).
It's based on an original research paper: The origins of phagocytosis and eukaryogenesis. The paper is open access and appears to be great reading, though it's conjectural and requires a little familiarity with fundamental biochemistry and cellular biology. Probably a good excuse to learn some of the details if you need to. These are topics that everybody ought to know about, even though our public educational system is way too inadequate to have done a good job of that.
Try reading at least the blog post, with a copy of Wikipedia close at hand.
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