I've been looking at a 10-22mm lens for my Canon T2i. I've also come across some other "wide angle" lenses, like the Sigma 24-70mm F2.8.
I don't understand how a wide angle lens can have such high numbers. The kit lens is 18-55mm; 18 is lower than 24 — so does that mean the kit lens is a "wide angle lens?
Traditionally, lenses wider than 24mm on full-frame are "ultra-wide". On a smaller-sensor Canon DSLR, a 15mm lens provides that same field of view (16mm on Nikon, Pentax, or Sony; 12mm on Olympus/Panasonic). So on an APS-C camera, a 20mm lens would be "wide" but not "ultra-wide" — but with the increased field of view of full frame, that would fall under ultra-wide.
On full frame, a 35mm lens is wide angle (but not ultra-wide); that same lens would be "normal" on an APS-C camera.
In either case, yes, the 18-55mm kit lens is a wide-angle lens. On a crop-factor DSLR, that nicely covers a reasonably wide angle, through normal, up to a moderate "portrait-length" telephoto.
(On some systems, some lenses designed for APS-C cameras won't work at all on full frame. Or, they'll work but with poor performance in the corners — and on Nikon, for example, this is compensated-for by automatically cropping-out the edges when a designed-for-APS-C lens is attached. But it's pretty much universal that full-frame lenses will work on APS-C.)
All of the lenses you mentioned are wide angle lenses; the distinction is that there are degrees of wide-angle. On your T2i, that 24–70mm lens at 24mm will be a moderate wide angle. The 10–22mm will be an ultra-wide (maybe ultra-ultra-wide?) at 10mm, and go to a more moderate wide angle at 22mm.
Mike Johnston wrote an amusing but opinionated article called Uses and Applications of 35mm lenses that attempts to break down the various categories of wide angle lenses. The focal lengths are for a full-frame camera, but you'll get the idea.
If you're looking to go really wide, Ken Rockwell’s article How to Use Ultra-Wide Lenses is a strong introduction to the significant challenges and rewards that you'll see from using ultra-wide lenses.
"Wide", "normal", and "telephoto" depend not just on the focal length, but on the format size as well. In the full-frame (35mm) world, any focal length shorter than 35mm or so counts as "wide angle" in that it captures a wider field of view than a "normal" 50mm lens (which roughly reproduces the field of view we see with our eyes).
In the large-format world (4" x 5"), a 50mm focal length is on the wide to ultra-wide side. In the APS-C world, that same focal length is slightly telephoto, and 35mm is "normal" (or close to it).
