There are several safety issues to be concerned about when playing Web-based games -- or, in fact viewing any Web page. But in general, Web games are harmless.
The issues involve some sites that insist on installing questionable software on your computer before you play the games. There are also Web-based games that are multi-player and bring you into contact with other real people, which is always a safety concern for children. In addition, some Web-based game sites require you to sign up for a membership, which could put your private information in peril. Here is how to deal with some of these safety issues.
Most free online game sites will require that you install either Flash, Shockwave or Java before playing. This is because the games require these as playback environments, just as Web pages themselves require a Web browser to be viewed. These three browser plug-ins are from reputable companies and you have nothing to fear from them. Flash and Shockwave and from Adobe. Java in from Sun.
Other sites may use more obscure plug-ins such as Wild Tangent, Unity or VirTools. These are also safe. If you ever have doubts about any of these, deny the request to install or upgrade the plug-in at the game site, and instead go right to the main site for the plug-in, such as Adobe.com, and install it from there.
Keep in mind that most plug-ins are actually "ActiveX" controls on Internet Explorer. This means you will get a request to install an ActiveX object, rather than a plug-in. However, a developer can build their own ActiveX object, which could be a game by itself, or a proprietary game playback engine. Or, it could be a piece of harmful software. When in doubt, search the Web for the name of the plug-on or ActiveX control and see what others are saying about it.
Some free online game sites not only ask you to install a game playback plug-in, but also require that you install a special browser toolbar or something else to gain access to their games. You should stay away from these sites. The toolbar or other software is usually designed to display advertising to you, even when you are not at that Web site. The worst case scenario with this software is that it is actually spyware, which means it could be gathering information about you and your computer and sending it back to the company that created it. A Web search for the name of the toolbar or software will tell you very quickly what others are saying.
If you are playing a multiplayer Web-based game, keep in mind that the person you are playing against could be anyone, and may not necessarily be the person they say they are. It is highly unlikely they can get any information from you unless you provide it to them during a chat. So be careful what you say. Keep the conversation about the game and not personal and you should be fine.