Cyclo-cross is cross country running on a bike. Usually on a 3-4 km circuit in parkland or woodland. Races last about an hour. Courses are mostly rideable but there are places where you have to run - obstacles, very steep hills.
It's great fun but physically quite demanding and you need to learn some new skills like dismounting and shouldering the bike.
Mud, and more mud.... traffic free circuit racing with a soft-ish crash landing etc.
Bikes: traditional cyclo X bikes usually have drop bars, cantilever brakes and often utilise bar end shifters for changing gear. A lot of people ride informal 'fun' events on hard tail mtbs though, or in fact any bike that will get round a lumpy muddy field without beating you up too much.
Went to watch a local club event once where there appeared to be a fairly even mix of men, women and children doing the course. It looked like it was great fun and a good workout on a winters morning.