This is a question I ask all new clients before I start a ride. A frightening number (well over half) get the answer wrong. Luckily they tend to get it wrong in the way that leads to skidding, rather than in the way that leads to a tumble over the handlebars. Even so, that's not great, as it means scuffed up trails and people who aren't stopping as quickly or effectively as they want/need to.
I asked this same question on Twitter and on Facebook. The Twitterati won: 79% of respondents said front brake. On Facebook it was a bit closer. Happily more people got it right (62%) than got it wrong (38%).
The main problem seems to stem from people being told that if they use the front brake, they will go over the handlebars. It is a fact that if you use just the front brake that is likely to happen (yes I've done it and experienced the consequences, as I'm sure most seasoned mountain bikers have at some point!). But, and this is an important but, it doesn't mean you should avoid using the front brake at all. As this short video clip illustrates, it is the one that does most of the stopping:
So, to summerise, to stop safely you need to use both brakes. The front one does 75% of the stopping, but if you want to stay on the bike and stop without skidding, you need to use both brakes!
In the UK your front brake lever is on the right side of the handlebars, and your back brake lever is on the left. If you're off to Europe for a summer jolly in the Alps, and hiring a bike, remember that they put the brakes on the other way round.