UniPlayer

It is possible to play Unity games offline or separated from the website they are hosted on. Offline is, of course, only an option if the game is not purely multi-player.

The good news is that you can download many Unity games and play them, regardless of an internet connection. Another benefit is that it is usually faster than playing the games in the browser.

Here are the steps to be done:

1. You need to install the Unity Web Player plugin on your system.
2. Download the latest version of UniPlayer from this website. It is the tool used to play Unity games.
3. Optional: Install a cache browser for easier access to Unity files. Firefox users can check out CacheViewer.

All steps above are one-time processes. To download and play a Unity game offline, do the following:

1. Open the game in your browser of choice. This downloads it to the cache of the browser.
2. Use a caching extension or browse your browser's cache manually to locate the .unity3d file in the cache.
3. Drag and drop the file into UniPlayer's interface.

NOTE: In Firefox you can also do [CTRL+SHIFT+I] to bring up the web inspector on the game page, click on the search box (left of the "Rules" tab), type in 'unity3d' and keep pressing [Enter] until you get a chunk of Javascript code. Double-click on the highlighted text and look through it for a file path that ends in ".unity3d", then just copy the path into the URL bar. If it doesn't include a domain, simply add the domain you're already on. (Example: http://thedomain.com/PATH/TO/GAME/FILE.unity3d). Load the URL and your browser should download the file.
In Chrome you can just right-click the page, click on "Show webpage's source code", do [CTRL+F] and search for 'unity3d'. You should then get an URL pointing to the file.