Put very simply, a MAC address is a unique identifier for your router that is used to make sure data reaches the right device.
Your broadband router has a unique MAC address which is used to identify your connection.
If you have Virgin Media broadband, this is the reason you can't use the router that's still at your place from the last occupants.
Your smartphone, laptop, games console... basically all connected devices have their own MAC address.
If you want a really deep dive on MAC addresses, Wikipedia has you covered.