Basics
The main goal of fencing is to hit your opponent within a certain set of rules before he does the same to you. Each hit counts for a 'touch', and the first person to reach a certain number of 'touch'es wins the bout (this number is usually five in pool competition, and fifteen in direct elimination). There are three different events in fencing, each of which has a different weapon and a slightly different set of rules. The three weapons are 'foil', 'epee', and 'saber'.
Collegiate competition:
Collegiate events at the conference level are team events. Each team has six squads -- split mens and womens, and one for each event, namely mens foil, womens foil, mens epee, womens epee, mens saber, and womens saber. Each squad has three starters, and as many backups as it would like. Since an individual can only fence on one squad, each collegiate team should have eighteen starters and as many backups as are available.
The team matches are divided up into nine bouts -- each of the three starters (or their backups) fences each of the other three fencers from the opposing squad. Each bout is five touches long. Whichever team has more bout wins wins the match. A squad may make substitutions at any time, as long as they give notice one bout ahead of when the substitution is supposed to be for.
USFA ratings:
The USFA gives out ratings in each of the three weapons for fencers based on their past performance. All fencers start off with a 'U' rating, for 'unrated' -- the majority of local fencers tend to be unrated. The ratings after consist of a letter from A-E, and a two digit number (such as D02). The number represents the year in which the letter was earned, and the letter is the relative strength of the fencer in that weapon. An 'A' typically means a nationally competitive fencer (and very strong locally), a 'C' usually means an experienced strong local fencer, and an 'E' means a stronger local fencer, with 'B's and 'D's being somewhere in the middle respectively. In order to have earned an 'E', a fencer has to have either won a small tournament, placed top three at a larger tournament (but without lots of strong people in it), or placed in the top 6 or 9 at a larger tournament with stronger fencers (see here for a ratings chart at the USFA site) Ratings expire after four years unless they are re-earned, and drop down one letter (for example, an A97 would turn into a B02 in 2002). While ratings are a good rule of thumb, they are not really an accurate measure of strength or a good predictor of how a bout will turn out. At the higher level, the USFA also gives out national rankings, based on points accumulated at national and international events.
USFA team competitions:
USFA team competitions are also gender split, weapon-based events, like in the collegiate competitions. The difference is that the matches are fenced to forty five touches, and are fenced relay. There are nine 'legs', which allows each of the three fencers on a team to fence every opponent once. Each bout stops when either team has a number of points equal to the leg number times five, with the last one being 9 * 5 = 45, and the points are cumulative. For example: fencer 1 from Team A is fencing fencer 4 from Team B in the first leg. The bout stops whenever somebody reaches 5 touches, and the next two fencers come on. The next two fencers fence until somebody reaches 10 touches. So, if fencer 1 'beats' fencer 4 from Team B 5-3, the next fencer from Team B needs to score 7 touches to finish the leg, whereas the fencer from Team A would only need 5 touches. Regardless of score, each leg can last a maximum of three minutes. The match ends when one team reaches 45 touches, or when time runs out. The ninth leg should start (barring time running out) with the score being 40 - X. Note that in this format, the strength of the last person is very important -- it's entirely possible to be down by an obscene amount and still win if your last fencer (the 'anchor') is sufficiently stronger than his opponent.