Within the EU, for EU nationals, there is a right to freedom of movement, which (slightly simplified) means you can visit, live and work in any other EU country. Depending on the country you go to, you can't always just rock up and stay there without doing anything, as there are things like Directive 2004/38/EC which means that in some countries, you may have to register. (There's also the whole tax thing that gets involved if you stay too long, but that's a different question!)
As an EU national, if I move to France, what are the rules about registering? Specifically, how long do I have to do it, where do I go, and what do I have to take with me when I go?