http://en.wikipedia.org/wiki/Progressive_rock
http://en.wikipedia.org/wiki/Progressive_metal
http://en.wikipedia.org/wiki/Progressive_country
Decent information throughout those articles, though that last one is basically worthless.
In my eyes, there are two types of progressive rock/metal bands - those which are progressive by style, and those which are progressive by ideal. To put it simply, the former includes bands that ripped off/built on Rush, and the latter includes bands that ripped off/built on King Crimson. I prefer the latter, personally... as these bands tend to far more closely represent the idea of being "progressive" as opposed to simply capitalizing on an already established sound.
As far as what being progressive actually means - most bands under the label feature elaborate arrangements, instrumental wizardry, complicated rhythmic structuring, unorthodox approaches to melody and harmony (dissonance isn't uncommon), use of atypical instruments, conceptual themes that span entire albums (whether lyrical, musical, or otherwise), elements of improvisation, heavy use of dynamics, and other things that "progress" outside of the boundaries that generally limit the music genre in question.
As far as what bands you can try out - most people will recommend stuff like Dream Theater, Porcupine Tree, The Flower Kings, IQ, Marillion, et al. These bands are fine, but I don't really think they're very progressive so much as they carry on the "prog" banner that was coined by bands like Rush, Yes, ELP, Gentle Giant, etc. Same old crap, new equipment and modernized sound.
I'd recommend bands like Hammers of Misfortune, Bubblemath, Estradasphere, Mr. Bungle, Gordian Knot, Sleeping People, Alchemist, Indukti, Devin Townsend, Pain of Salvation, Subterranean Masquerade, Maudlin of the Well, Dysrhythmia, Upsilon Acrux...
However, I imagine that most progheads would disagree with me regarding the "prog" and "progressive" dichotomy. Regardless, there's a lot out there on both ends of the spectrum. Dive in.