Time of possession is defined quite simply as the amount of time a team is in possession of the ball during a timed sporting event. It's highly relevant in games like American football, soccer, hockey, and basketball, in which scoring is almost always accomplished while a team is formally in possession of the ball. The exception to the rule of course is defensive scoring, which can happen in the form of a defensive touchdown (in American football) or an own goal (in soccer). Ignoring the usually minor effects of defensive scoring, time of possession often directly correlates with the number of points a team scores during a timed game. If you don't let the other team get the ball, they'll score less and you'll score more!
It's interesting to think about the strength of the correlation between time of possession and scoring across various sports, as it is most definitely not constant. American football is a great example of a sport in which the correlation between time of possession and score is extremely weak. Because different kinds of plays affect the clock in different ways, a team's playing style will dictate its average time of possession, regardless of how good it is at executing that style (only to a point, of course). In addition, the amount of time a great play takes in football is usually only a few seconds longer than a mediocre or bad play, so time and skill aren't terribly related.
Consider the stats this year. The Baltimore Ravens led the league in time of possession, no doubt to give their dominating defense rest. But the delightfully mediocre Houston Texans (8-8) trailed the Ravens in total TOP by only a minute and a half. In the context of sixteen sixty-minute football games that seems like nothing, but it's a testament to how many big, quick plays the Texans gave away! In fact, it's a good rule of thumb across many sports that a team that has a good number in the TOP column but a bad record tends to give away big plays.
Despite the weak correlation in football between TOP and record, NFL statisticians continue to track it. Why?! Hockey has been due to keep track of TOP for years, although the NHL doesn't yet. Think about it: how often do you see hockey breakaways compared to, say, fast breaks in basketball? It's not a mass-media conspiracy; hockey is a battle of attrition and game outcome has a great deal to do with the time each team holds onto the puck.