Unfortunately, my flight was very late, so I didn't actually arrive at Nice airport until just before 1am. The first thing to note is that the N100 nightbus is very prompt about leaving, and at 30 seconds past 1am you'll see its rear lights disappearing off into the distance...
The bus stop for the N4 nightbus is not well signposted from the airport, and the route is a bit complicated by various carparks and roads that don't seem to show properly on maps. Don't expect to be able to find it very quickly your first time trying! However, it is more frequent than the N100, so may work better.
Generally though, the bus service is pretty good in the day and early evening, and not too bad in the late evening, it's just around midnight when things start to get hard, and you'll probably need a delayed late flight (like mine...) before you'll end up with issues.
At 1am, there was a huge line of taxis available at the airport ready to whisk people into town. When I returned at 1:20 (having given up trying to find the N4 bus stop in the dark), there was a short wait for a taxi to arrive. At that time of night, taxis are a flat rate of €35 to the centre of Nice. I believe there are different flat rate prices for other areas. (uncovery's answer has a link to a slightly outdated price list)
Oh, and one thing to note for coming back from Nice to the airport - at the moment (2013) there's lots of work going on around the station, and Place Massena. Lots of bus stops have been temporarily moved, but there are maps around which show the new locations. The #99 bus (airport <-> station) and #98 bus (airport <-> city by coast, runs later) are both very good and worth getting, but allow a bit of extra time to find the stops!