The most basic explanation is that under tournament settings (things like 4-stock, 8 minute timer, no items, legal stages), a character higher on the tier list is more likely to win a tournament than a character lower on the tier list.
Here's an example for Brawl: Meta Knight is the top of the tier list; he's actually in his own separated tier because he's just that much better than the characters directly below him (Olimar, Diddy, Ice Climbers, etc). And you know what happened at the most recent international tournament, APEX 2012? 1st place: Meta Knight. Second place: Olimar. Third place: Meta Knight. Fourth place: Pikachu/Ice Climbers. All of these characters except for Pikachu are currently in the top 4 of the entire cast, and Pikachu is 8th and ESAM is just ridiculously good. Doubles? The Grand Finals match was two teams of two Meta Knights each.
Same tournament but Melee: All the top-placing players were using characters from Fox through Captain Falcon on the tier list, as I recall. The only reason Young Link won was because Armada pulled him out for one set after playing Peach the entire rest of the tournament because YLink is considered a low-tier counter to Jigglypuff.

I would like to announce the opening of the Smash Flash Back Room. Like the Back Room groups for the official smash games, the Smash Flash Back Room will consist of a number of highly skilled players discussing the competitive aspects of SSF2: advanced techniques, stage legality, technique legality, and tier lists, among other things. This separate forum will have separate forum topics for separate discussions for increased clarity and efficiency. Also, it is essential this project that the discussions and results are held to the highest standard of quality possible. To this end, in order to become a member of the Back Room, you will have to prove that you are an intelligent, knowledgeable player through a brief application process. Things like experience in official Smash titles (especially tournament experience), ability to communicate opinions, and overall knowledge of SSF2 are all plusses in this process. If you would be interested in applying to become a member of this new project, you can go to http://smashflashbackroom.englishboards.com/forum.htm and fill out an application. Some personal invitations have already been sent out through PM; however, if you didn't get one that doesn't mean you won't get in: it just means I probably missed you when I was sending out my PMs. Good luck everyone!