Before you can install it on your phone you will need to make sure that third-party apps are allowed on your device. Go to Menu > Settings > Security > and check "Unknown Sources" to allow your phone to install apps from sources other than the Google Play Store.
Once you have downloaded the file on your computer, connect your device to it. At this point, you may be asked if you simply want to charge your phone or connect it as a "media device." Choose "media device". Then, simply find your phone's folder on your computer (it will be in "My Computer" or "Computer" on Windows machines) and copy the APK file into a folder of your choice in your Android smartphone.
You will now be able to search for the file location in the "my files" folder on your device. Find the APK file, tap it, then tap "install". And you're done.
You can also install APK files from the browser on your Android smartphone or tablet. Just open your browser, find the APK file you want to download, and tap it - you should then be able to see it downloading on the top bar of your device. Once it's downloaded, open your "downloads", tap on the .apk file, and tap "yes" when prompted. The app will begin installing on your device. Simple.