The first thing to check is that the volume is turned up on the device. The volume controls on the left-hand-side of the device can be used to increase and decrease the volume, so try adjusting these this whilst a video is playing to see if it affects the volume.
Another way to adjust the volume is to access the iOS control centre tray, which you can slide up at any time by running your finger up onto the screen from the home button. Here you will see the current volume level and can drag it right to increase it.
If the video is still mute then we need to try resetting your device’s factory settings. This will correct any problems linked to Apple's media player. Please note: this will mean losing your personalised wallpaper or any other customisations you have made to the settings but your personal data and apps will be safe. If you don't mind giving this a try then you can find the setting under the iOS Settings > General > Reset > Reset All Settings.
Do not confuse this with ‘Erase all Content and Settings’ which will delete everything from your device! If all else fails, try taking your device to an Apple store.
One other outside possibility is that your device is transmitting to an Air Play device. This will only be possible if you own an Apple TV or similar device. You can enable / disable transmission to an Air Play device via the system control panel tray pictured above.