The main browser plug-in used for web-based games is Flash. If you just bought a new computer, chances are flash came preinstalled on your machine. however, it may not be the most recent version.
Other plug-ins, like Shockwave, may or may not come installed with a new machine. And they may also be an older version.
So the first thing you may want to do before trying to play some free online games is to make sure you have an up-to-date version of both Flash and Shockwave.
You can do this by going to the Adobe website. Here is where to check for Flash. Here's where to check for Shockwave.
You also may want to check to make sure you have a recent version of Java. You can find that at Sun's website.
There are actually two forms of web browser plug-ins. For all browsers except for Internet Explorer, plug-ins come in the same form. For instance, the same browser plug-in that works for Safari on Mac, works for Firefox on Mac. However, on Windows, while there is also a Firefox plug-in, you need something different for Internet Explorer. What you need is an ActiveX control. ActiveX is a special type of browser plug-in architecture used only by Internet Explorer in Windows.
One advantage of ActiveX, is that the installation procedure is much simpler. In fact, an ActiveX control can be automatically installed the first time you visit a page that requires it. So if the user without Flash, visits a page with a Flash movie on it, and they are using Internet Explorer, they will get flash and automatically be shown your content without having to leave the page.
Browser plug-ins, on the other hand, may require the user to download installer and run the installer manually. However, if all they require is an update of their browser plug-in, the plug-in may be able to do that without requiring the user to do anything additional.
Some plug-ins, like Shockwave, even have plug-ins themselves. For instance, the bare minimum Shockwave install does not include commonly use components such as 3-D and physics. But, the first time the user visits a game that uses these, Shockwave will update itself, adding these components.