What's the difference?
I had this one originally planned as a typical student error until I realised even natives get this wrong too. So...
Barbecue (British English)
This is a piece of equipment (as shown in the picture above) that we use when we want to cook something outside, usually over some wood coals. If we have a barbecue, we cook food outside; if we barbecue something, particularly meat and burgers, then we cook it on a barbecue.
Barbeque (American English)
Firstly, this word is simply a different spelling and can be used as the same meaning as British English, but it is also used to define food that is cooked over wood coals and cooked slowly in some kind of covered container and may even be flavoured by some kind of… barbeque sauce. But read on...
Grill (American English)
Many Americans prefer this definition to the more British English definition of barbecue. It is defined as a flat frame of metal bars on which food can be cooked over a fire. Indeed, it is this definition that is often used by English language students.
Grill (British English)
For the Brits, this really has very little to do with cooking outside. The definition here is a part of the oven (or stove in AmE) which produces strong heat to cook food that has been placed underneath it.
Bar-B-Q This is simply an abbreviated form of the American English Barbeque, which carries both British and American meanings.
Barbie Yes, as well as being Ken's girlfriend in the dolly world, it is the informal Australian and British English word for barbecue.