What do those designations mean?

[Owlie]

It usually has to do with the amount of travel and differences in geometry. A trail bike, or all mountain is going to have more travel (5" or so ) than an XC bike (3-4") and will be more slack in the geometry. More slack is less aggressive/forward in geometry; the "sit up and beg" verses 'XC race" style. I have heard "sport".... I have not a clue but it makes me think of a Sports Authority style box store bike.

As far as I can tell, the "sport" is a more "general public-friendly" bike. It'll do a little of everything, but not be very good at it. The "sport" series from at least one particular manufacturer appears more upright too.