"Gauge" means thickness. The higher the number, the thinner the wire - so 28g wire is very thin, 16g wire is pretty thick.
Here's a link to a great article at Maille Artisans that talks about gauge AND it has a useful chart. http://www.mailleartisans.org/articles/articledisplay.cgi?key=221
Note the part that talks about wire manufacturer's tolerances. It is quite possible that two different batches of the same sized wire from the same supplier will actually be different thicknesses. If you're buying jump rings and this is a concern to you, take a look at Urban Maille (www.urbanmaille.com). There are definitely cheaper sources out there for jumprings but Urban Maille is the only one I know that GUARANTEES that their ring thickness are consistent.
As if the difference between AWG and SWG wasn't complicated enough, many sites don't mention which rating system they're using. To make it even more worse, some sites (like The Ring Lord) use SWG for 18g and larger and AWG for 20g and thinner!!!!! If in doubt, ASK and don't assume!!!
For earwires, you want 20, 21 or 22 awg. Anything much thicker or thinner than that is usually uncomfortable to wear. Other than that, there are no particular "rules" about what type of wire to use, it really depends on the look that you're going for. It's hard to get a substantial "chunky" look with 28g wire; conversely, 10g does not lend itself to light, airy-fairy designs.
Half hard, full hard, dead soft etc. refer to the "temper" of the wire - i.e. how hard it is. What you should be buying really depends upon your personal preference and the kind of work that you're doing. Complicated wrapping a la Eni Oken is going to call for dead soft wire; other applications will be fine using half-hard. Personally, I'm not a fan of full hard wire because I find it too springy but I may be alone in that view. FWIW, I think it's MUCH easier to harden wire than anneal (soften) it, so if in doubt, I'd advise that you buy the softer stuff. Generally speaking, I use dead soft for 18g and thicker; half-hard for anything thinner.
If you're buying by weight - some places sell by the avoirdupois ounce (i.e. a regular ounce that has 16 to the pound); others sell by the troy ounce which is heavier than an avoirdupois ounce (I *think* there are 14 troy ounces to the lb but I could be wrong on that). Anyway, you just need to know these if you're trying to do some comparison shopping.
And then of course, there's the fact that wire comes in different shapes - round, half-round, square. Half round can be very useful for wrapping because having a flat surface against the object being wrapped can make the whole process go easier. Other than that, shape is largely a matter of aesthetics so you can use whichever one you want.
Finally, May 2006 saw silver trading for the highest prices in 24 years (aaargh!). If you're a beginner, you may want to consider practicing on something a little less expensive, such as copper.
Have I helped or merely added to your confusion? I hope it's the former!!!!