Off-road cycle racing, dating back well before the invention of the mountain bike. Originated in Europe as winter knockabout training for road racers in the 1940s or so, but developed as a specialist discipline since then. It is particularly popular in small pockets in Europe, notably Flanders (with 6 of the world's top 10 riders and about 90% of the TV coverage) and the southern Netherlands (with 1 of the top 1 ...), the Nord and Brittany in France, the Basque country, northern Switzerland and both halves of the former Czechoslovakia. Riders from the USA - mainly New England and California - have shown promise in the last couple of years.
Races are generally held on circuits around 2 km in length, covering a mixture of asphalt, path, grassland, mud, sand and anything else the organisers can find, including artificial obstacles or sections which are unrideable (flights of steps are not uncommon) or rideable only with extreme difficulty, forcing riders to dismount and remount repeatedly; doing so whilst wasting a minimum of speed and energy is one of the primary skills of a good cross rider. Modern courses are faster than they used to be, as the sport has sought to distinguish itself from the slog of mountain bike racing. Races are fairly short - usually 50 minutes to an hour duration for senior events.
Race tactics are pretty straightforward, by comparison with road racing; start flat out in order to get to the narrow parts of the course at the front, and keep going as hard as you can as long as you can. There is little opportunity for meaningful teamwork unless there are long road sections, although that doesn't stop a lot of bitching when Belgians from different trade teams get to ride together in the national squad and allow commercial loyalties to outweigh national ones.
The season lasts from September to early February, building up to the World Championships (decided by a single event in each category) at the end. The other major international events are the UCI World Cup series (currently 6 races in different countries) and the Superprestige series (mainly in the Benelux coutries).
A cross bike is broadly speaking the same shape as a road racing bike, but with wide clearances (and no chainstay bridge) to prevent it from clogging up with mud, cantilever brakes (same reason) and narrow knobbly tyres. Gear levers were traditionally mounted on the bar ends, but now mainly gear/brake levers of the Campagnolo Ergopower or Shimano STI type are used; cable routing is often non-standard to keep it out of the mud and to make the bike easier to carry. Unlike mountain-bike racing, the riders are allowed to receive technical assistance, usually in designated pit areas; bike changes are common, especially on muddier courses where the pit crews can provide clean bikes every lap if necessary.
UK-specific bit: In Britain cyclo-cross is one of the most accessible forms of bike racing; most events (other than those at international level) can be entered on the line with no licensing or membership requirements, and more or less any machine is acceptable - plenty of people ride mountain bikes. Every promotion also includes a short race for under-12s on a simplified course.