Q: Is this my battery or alternator?

This may be a sign of a weak battery, a failing voltage regulator or failing alternator. As you may know, the alternator is responsible for charging the battery while the engine runs by generating amperage and recharging the battery while the engine turns the alternator. The voltage regulator is a unit that regulates the charging of the battery by the alternator and maintaining a certain amperage while the engine runs. When the alternator is not working properly, this may result in the battery quickly losing charge and the car losing all power as you have described. I would recommend having an expert from YourMechanic come to your location to diagnose and repair your charging system.