Sounds like it's a software issue. Your phone might have gotten a virus. Yes, people make viruses for Androids. Sad, I know.
Back up anything you need to save to the SD card, then turn the phone off and remove the SD card.
Next, hold the volume down + power buttons for approximately 30 seconds or until the HBOOT menu appears. Try volume up + power if the first doesn't work.
In the HBOOT menu (HBOOT is essentially BIOS for smartphones) select the "restore to factory defaults" option using the volume keys to navigate, and the power button to select it.
This will delete all apps and data on the phone, and return it to the stock settings, the condition in which you bought the device.
Re-insert the SD card and move the media you backed up back to the phone. If the issue persists, take it to a cell phone repair shop and have them take a look at it.