A lot of DVDs I have say "Mastered in High Definition", does that mean it is an HD-DVD?

No, it just means that the transfer is mastered in HD, which is then downconverted to SD.

ohh, gotcha, thanks

If it doesn't have the peculiar red case, it's not an HD DVD.
If it doesn't have the peculiar blue case, it's not a Blu-ray.

Neijii. Movies tend to be transfered from film to digital at higher resolution. That way when they get encoded the encoder does some downscaling at the same time and can pick how to make it better (i.e. it has a bit more info to try and decide what to do) so good DVDs went Film->HD ->SD for DVD. BD and HD DVD they try to work with 4x masters

as for what Icemage I will add
the red cases play in HD DVD players, the blue in BD players and neither HD DVDs nor BDs play in DVD players*

Some HD DVDs are combos, the combo is a disk with an HD DVD on one side and DVD on the other. The DVD side that is normal DVD and not HD will play in the DVD player