you can look at HD in two ways really. Firstly HD refers to certain resolutions which are higher definition than standard TV resolution. The main HD resolutions are 720 and 1080 and are so called because of the vertical resolution of the signal. So 720 would be 1280 x 720 and 1080 would be 1920 x 1080. From there you also have whether an HD signal is progressive scan or interlaced, with progressive scan being better. So you can get 720p/i or 1080p/i.
So for a screen to support one of these HD resolutions it must have a vertical resolution which is large enough to support the signal. So a 1280 x 1024 can support 720p/i in it's normal form, but a 1080 source would be downscaled a bit. A 24" monitor though with 1920 x 1200 resolution could support a full 1080 HD source though.
After this, the other thing to consider for a truly "HD ready" screen is whether it supports HDCP which is basically an encyption method. This will eventually be incorporated into HD sources like HD-DVD's, Blu-Ray etc, as well as in HD hardware like DVD players. If these feature HDCP encryption then you also need a screen which support hDCP over it's digital (either DVI or HDMI) interface. HDCP is far from being widely used though and no doubt there will be some cunning ways round it at some point with decrypter boxes or the likes.