While the National Hockey League is the best hockey league in the world, there are literally hundreds, maybe thousands of other leagues of various skill. Here is a partial list of the other leagues around the world, including only the best and most well-known.

North American

Professional

American Hockey League (AHL)
The AHL is the top minor league for the NHL. Most of the players that play in the AHL are either past or future NHL players, although there are few career AHLers. The AHL is to AAA in baseball, in that a team's top prospects are usually playing in the AHL.
Wikipedia, Official Site

East Coast Hockey League (ECHL)
The ECHL is a tier below the AHL; the AA affilate of NHL teams. Few players other than enforcers or goalies come out of the ECHL to play in the NHL, although there is a lot of movement between the ECHL and AHL.
Wikipedia, Official Site

Amateur

Canadian Hockey League
The Canadian Hockey League is the governing body of the three following major junior leagues in Canada. Major junior is the top junior level in Canada; there are dozens of other junior leagues, but most future NHLers come from major juniors. The NCAA considers major juniors to be a professional league because the players receive a stipend.
Wikipedia, Official Site

Ontario Hockey League (OHL)
The OHL is known for producing the highest talent of the three major junior leagues, and is the most prominent and followed of the three leagues.
Wikipedia, Official Site

Quebec Major Junior Hockey League (QMJHL)
Located in Quebec, the QMJHL has slipped in esteem recently, but has a history of producing some of the best goalie prospects.
Wikipedia, Official Site

Western Hockey League (WHL)
The most defensive-minded of the three major junior leagues, the WHL is mainly located in western Canada and parts of the US.
Wikipedia, Official Site

NCAA Hockey
Many American players go the college hockey route, playing Division I hockey at various universities. Much like in college basketball, Div I hockey has a tournament in March to declare the college champions; the final four teams in the tournament participate in the "Frozen Four."
Wikipedia, Official Site

United States Hockey League (USHL)
The USHL is considered a Junior A league, which is a tier below major juniors. The reason for inclusion is because most players that want to play in the NCAA instead of major juniors play in the USHL, as it does not affect their amateur status.
Wikipedia, Official Site

European

Czech Etraliga
Czech Republic. Wikipedia, Official Site

SM-liiga
Finland. Wikipedia, Official Site

German Ice Hockey League (DEL)
Germany. Wikipedia, Official Site

GET-ligaen
Norway. Wikipedia, Official Site

Kontinental Hockey League (KHL)
Russia; the second best league in the world. Wikipedia, Official Site

Slovak Extraliga
Slovakia. Wikipedia, Official Site

Elitserien, colloquially Swedish Elite League (SEL)
Sweden. Wikipedia, Official Site

National League A (NLA)
Switzerland. Wikipedia, Official Site