My daughter, who is deaf in one ear, likes to watch movies and listen to music on her iPad. On long road trips or while she's in the same room with her little brother, she uses headphones. But she uses them a bit differently than most.
Instead of playing audio to both sides of her headphones, we set it up so that all audio is redirected to her left ear. Both iOS and Android let you adjust the audio volume balance between the left and right channels as well as enable mono audio to ensure no sounds get lost in the process.
iOS audio balance
In iOS 9.3, head to Settings > General > Accessibility. Scroll down to the Hearing section and you'll see a slider for left/right balance. Move the slider to the side where you would like to increase the volume level.
For many audio sources, stereo sound features different sounds and levels in the left and right channels. With mono sound, you'll hear the same sounds in both channels. Above the volume balance slider on the Accessibility page in Settings is a toggle switch to enable Mono Audio.
Android audio balance
You'll find these audio settings in a similar place on Android. On Android 4.4 KitKat and newer, go to Settings and on the Device tab, tap Accessibility. Under the Hearing header, tap Sound balance to adjust the left/right volume balance. Below that setting is a box you can tap to check to enable Mono audio.
For owners of the Samsung Galaxy S4 (or later), a setting called Adapt Sound gives you even greater control of your audio output. Not every music app supports the feature, but it does work with Samsung's built-in music app and Google Play Music.
To set up Adapt Sound, put your headphones on and find Adapt Sound in the Sound section of Settings. It will play a series of beeps and sounds at different frequencies, similar to a hearing screening you might have experienced in school growing up. You answer whether you can hear each sound, and the app will then adjust the strength of certain tones and frequencies to help you in areas where you may be deficient.
For my daughter, we enabled mono audio and moved the balance slider all the way to the left, which is her good ear. If you move the volume slider to one side without selecting mono, you lose any audio that is specific to the other side.
A good way to check your volume balance settings is playing the song "Money" by Pink Floyd. In the beginning of the song, the cash register sounds pan back and forth between the left and right channels.