The Entertainment Software Ratings Board (ESRB) is a group of people who play games before they are released and give them a rating to inform consumers as to what age the game is appropriate for. These ratings are plainly stated on the game's cover.
For those unfamiliar to the rating system, the game ratings roughly correspond to the movie ratings with which most people are rather familiar.
E is for Everyone. This is given to a game that is appropriate for all ages and generally contains nothing more sinister than "comic mischief". This could mean jumping on enemies' heads to defeat them, or some light beat-'em-up action where no one really gets hurt. The E rating is something like G or PG for movies.
T is for Teen. These games usually have more violence in them, but not much in the way of blood or gore. It can also have mild language or sexuality, and maybe some drug or alcohol use. It's like a PG-13 movie.
M is for Mature. These games are meant for adults. They contain some mixture of violence, sex, drugs, or language. This is like an R rating.
There are a couple of other ratings such as EC (Early Childhood) and AO (Adults Only) that are seen on a handful of games, but are not very common, and pretty self-explanatory.
If you are unsure about a rating, or why a specific game has a certain rating, you can check the back of the box. There is a black and white box that contains a brief breakdown of why the ESRB gave it that rating.
Keep in mind that just because a game is rated E for being nonviolent, it does not mean it is necessarily a children's game. Most puzzle games are rated E, and some border on being too complex for children.
Although there are a few flaws in the rating system (such as the ESRB not playing the full game, and often not the final product, or the board not necessarily being made up of people who are particularly well informed about gaming culture), it holds up well when followed. Ratings are fairly consistent over the spectrum, and so can be used without many problems. Like movies, however, it is a system that only works if enforced by the guardians.