If you mean blu-ray drive. then a blu-ray drive with a blu-ray disc plus an HDMI cable to a suitable screen should play fine since there is a rights-managed path from end to end.
If you mean you have a DVD drive, then a blu-ray disc won't work in it.
To give a little perspective on the "will my computer play HD" thing, whilst 1080p is quite recent to TV screens, resolutions way beyond 1080p have been normal on computers for many years. However one recent benefit is that large-screen monitors have suddenly become cheap as chips. You should be able to play a 1080p video source with no hiccups at all.