If you already have the apk file
If the file is downloaded directly on your Android device, then you can navigate to the file location and tap on it to to install. During the installation follow on screen directions to complete setup. During installation, Android OS displays all the permissions that the app is requesting on the device, say access to camera, access to location, access to internet etc. Review these and follow the onscreen directions to complete installation. Why a device needs different permissions is usually listed on the developers website as well as the play store.
If the file is downloaded on a pc or different device, you first have to transfer it and then install it. File can be transferred to the device in several ways, bluetooth, infrared, otg drive, memory card or as an email attachment. In some devices, I have noticed that a .apk file is not downloaded. This is probably a security measure by either Android OS or an antivirus program. The workaround I used is to rename the file to some random extension(say jpg), transfer it to the device, then rename it back to .apk and then install.
Before installing an application using apk file, the "Unknown sources" checkbox in Android settings should be checked. Goto Settings>>Security, scroll down to find "Unknown sources" and check the box on the right. This is unchecked by default and unless this is checked, Android does not allow installation using an apk file.
If you have an antivirus installed, the antivirus also may have a setting restricting the installation of such files. Check your antivirus(or disable it temporarily if possible) for settings similar to 'unknown sources', 'third party apps', 'install using apk' files etc. I use AVG and when 'Unknown sources' is checked, it prompts me that a threat is found in settings but does not restrict installation. AVG also has an option to ignore the threat.