FIFA is the world's governing body over all leagues that are affilliated. Below that are regional governing bodies--the US belongs to CONCACAF, which stands for "Confederation of North, Central American and Caribbean Association Football". These regional bodies run tournaments at both club and country level--to determine the best club team in CONCACAF there is the CONCACAF Champions League. However, to determine the best country in that region, there is the Gold Cup. Each region had their own version, and each region determines how often these tournaments are held.
Below that there are individual leagues in every country--some have multiple levels; i.e. a major league and perhaps several minor leagues. There are a few inter-continental or inter-league tournaments (The Champions League, Club World Cup, Cop Libertadores, CONCACAF Champions League, etc).
With regards to MLS, in particular, there are currently 18 teams (2 will be new to MLS in 2011, but both have been in existence in a 'lower' league prior). Unlike most all other countries, MLS favours a 'conference' approach, rather than a single table--it seems to be an easier way for many ill-informed sports fans in the US to understand soccer better, but it is revolting to purists who feel there should only be a single table. This is mainly due to the fact that the conference system lends itself to play-offs, which is not common (as we understand it) in most other leagues. The simple fact in most other places is that if your club finishes the season with the most amount of points (from wins and ties), your team is the champion. There is no real winner take-all scenario.
Back on track though--check out the MLS website to see what teams are in the Eastern and Western conferences, and it all may make a bit more sense. Also, at present, there is no system of promotion and relegation in MLS as there is in other countries' leagues. Think about how interesting it would be if the teams that finished worst in the league during a season (maybe even the bottom three), get relegated to playing in a lower division until they can earn promotion back to the top flight. I'm not fully versed in baseball, but perhaps that would be the most comprehensible American version of this--the Cubs finish dead last in the entire league....well, next season they play in AAA. Whatever AAA team won their league gets promoted. It always continues like this. I hope that makes some sense.
I'm headed out for the night now, but I can write more later, if you've followed and/or are interested in knowing more. Mind you though, you can do plenty of research on your own even based on what I've just talked about. But let me know if there are specific questions and I will do my best to answer them.
As for the Fire though, Fire confidential is probably the most knowledgeable and accurate site--Guillermo, the guy who runs it is good people and will answer questions as best he can.
Setion 8 Chicago's forums are also a good resource, but you'll have to put up with folks like me all the time....not picnic.