Download .apk file. Any android app outside of Google Play come in .apk format.
After the download is complete, tap on the download completed notification to install. In this example, the user is trying to install Vidmate.apk.
If you are installing an app outside of Play Store for the first time, you may see an installed blocked message. Just tap on settings.
For older versions of Android, you will see something like this after you tap on settings in the above screenshot:
For newer versions of Android, tap on settings and allow the installation from browser/source you just downloaded the app. In my case, Chrome.
Go back to the installation screen and you will see an install window. Tap on install and you are ready to explore the app after it gets installed.