"barbecue" and "grill" are generally done by open fire, and they mean just about the same. However, "grill" sometimes are done indoor over a range that has a grill function. In US, range consists of "oven" (for baking) and "stove" (for cooking). However, some ranges have a grill plate in the middle of the stove (usually has 4 burners on top of the range).
"bake", "roast", and "broil" are generally done in a range (the oven portion). "Bake" is for pastries, such as cake, bread, cookie, pizza, ... and so on.
"roast" is for meat, such as turkey, beef, ... and so on.
"broil" is when in the oven you turn on the burner that is on top of the oven,
you can use this burner to blacken (or darken) the surface
of whatever you are baking or roasting,
"toast" is generally done in a small appliance like "toaster" or "toaster over".
basically, it is to warm or to crisp the bread (or other pastries).