Schools usually block computer games in order to encourage students to work while in their computer labs, to prevent the spread of viruses, as well as to prevent the spread of inappropriate material. All schools employ someone to monitor Internet use, as well as to keep the school's Internet functioning. If you are careful, it is possible to play nonharmful games on school computers.
Most schools will respond quickly if a particular game becomes popular, and will block the site that allows access to it. By using a meta-site, however, it is possible to access a variety of games. Sites such as "funnystuff.com" or "i-am-bored.com" all have a good range of games. Be careful, however, as many web blockers will have certain URL keywords banned, such as "games." Some newspapers, such as the UK Guardian "guardian.co.uk" has a section on online games which can be played.
This is a major concern of the school in blocking certain sites. Some gaming sites can spread viruses, particularly if file-sharing or downloading is taking place. This can be true even of sites that require a flash download to play the game. In general, find a game that can be played without downloading anything and you will avoid getting a virus on the school's computer. Viruses can be traced, and schools take tough action against this.
Even if you are not looking at inappropriate material directly, many gaming sites have material unsuitable for minors on them. This may cause the site to be blocked automatically by a school. By finding a portal through a respected site, such as "guardian.co.uk" it will be possible to avoid contact with inappropriate material, whether you were looking for it or not.