Google Play won't let you install apps to your device if it is deemed 'incompatible' due to screen size, while others apps are made available only for specific handsets. In both cases, you may be able to get around that through a manual installation, a process often referred to as "sideloading". Here's how to do it.
Technology weblog Make Use Of outlines the relatively simple steps involved in sideloading apps to your Android device. The most difficult stage is usually getting hold of a copy of the APK file for the app you wish to install. These can often be found on forums or developer sites, but be cautious; installing an unknown APK file poses a clear and definite security risk.
Once you've obtained the file you need to connect your phone or tablet to your computer via USB, turn on USB storage mode and copy the APK to your device. Go into Settings>Security and enable installation of apps from unknown sources. Once you've done that fire up a file browser (such as ASTRO), locate the APK file, and open it. Your device should then recognise the file and your app should be running in a few seconds.
You shouldn't use this approach for piracy, and it's worth emphasising again that installing apps from unknown sources is risky. However, for apps that have been removed from Google Play but that you still find useful, it can be a handy alternative.