Sparta (http://www.sparta.cz) are the richest and most successful Czech club, and income from competing in the European Champions League has enable them to develop their compact ground into a 20,000-capacity all-seater stadium.
Slavia (http://www.slavia.cz) are the city's second club - Sparta's traditional slightly-poorer city rivals, although they've often given them a run for their money over the years.
Bohemians (http://www.fc-bohemians.cz) were the city's traditional 'other team', but acheived fame in Britain following their performance against the Ipswich Town side that won the UEFA Cup in 1981, and reached the semi finals of in 1983.
In recent years, though, Bohemians have been eclipsed by Viktoria Žižkov (http://www.fkviktoriazizkov.cz), a traditionally lower-league outfit that achieved success after the Velvet Revolution - they reached the first division in in 1993, and have stayed there ever since, improving year on year. In 2002, they knocked Glasgow Rangers out of the UEFA Cup.
Dukla are the most interesting of all. They were one of the 'crack Eastern European outfits' fabled in the 1950s and 60s, competing in friendlies across Europe, and they reached the Semi-Finals of the European Cup in 1967 and the Quarter-Finals of the UEFA Cup in 1979. They gained cult status in the 1980s through the Half Man Half Biscuit song 'All I Want for Christmas is a Dukla Prague Away Kit (although in fact, they never actually had an away kit - but demand for them was such that a manufacturer of retro football shirts decided to make one in the 1990s, and it's proved very popular). However, away from the Western myth-making, Dukla's success was largely based on being favoured by the Communist authorities, as they were the Army's chosen club. In the post-Communist era, support for Dukla dwindled because of their historic connection to Communist repression, and they hit financial trouble after the Army set them free in 1994, and were relegated to the Czech second division. This was followed in quick succession by relegation to the third - and by this time, Dukla were on the brink of extinction as they couldn't afford to pay their registration fee for the next season. However, they avoided both fates by merging with second division Pribram, 50 miles south-west of Prague, who were in the second division at the time, and changing the team name to Dukla Pribram. In 1999, the club was renamed Marila Pribram, and in that form they have returned to the Czech first division, and played Aston Villa in the UEFA Cup.
There's a great little football souvenir shop in Prague in the Nove Mesto, where you can buy shirts, scarves and glasses of all Prague teams, including some old Dukla stuff. I'll add the address when I've found out what it is.