It's a form of cross-country running in which the competitors have to navigate round a course using a map and compass. The course consists of a set of control points that the competitors have to visit in sequence. Races usually take the form of a time-trial so that competitors can't just follow each other.
At the top end of the sport, the competitors cover rough terrain at speeds worthy of the best cross-country runners using just subtle land form and vegetation features to navigate by. Younger runners and beginners compete on courses that require simpler navigation, following paths and other simple features, then gradually move up to technically and physically harder courses as they gain experience and confidence. The maps are very detailed, usually at 1:10,000 or 1:15,000 scale with a 5 metre contour interval. Whoever covers the course fastest wins. A device at each control point is "punched" to prove that the runner has passed through that point; until recently the device would have been a simple pin punch, but electronic punching systems are increasingly common. The competitors are free to choose their route for the "leg" from one control point to the next, so as well as being tested for their running speed and map-reading skills they often need to make tricky route choices while in oxygen debt. Over a long leg of a couple of kilometres, making the right route choice can easily save minutes.
Where can you go orienteering in France?
Most (but not all) departments in France have at least one club to organise orienteering events: big clubs might put on 5 or 10 races each year, while the smallest struggle to hold 1 or 2. Many areas also have ESOs (espaces de sports d'orientation) where maps of permanent courses are on sale (generally through a town hall or a tourist office) allowing you to turn up and have a go whenever you want. It's a small sport in France, with just five or six thousand FFCO members (licenciés), so newcomers are generally made very welcome.
The FFCO (French orienteering federation) web site includes links to clubs in all the various regions and departments.