Here are some comments from Wombat <firstname.lastname@example.org>...
Well, I showed up for a class/meeting with the #10 needle I thought I would need, only to discover I was supposed to have a #10 sharp and I had a #10 crewel. A #28 tapestry did suffice and I then went home and did research.
Eyes. The choices run from round to oval to long or short oval. Round eyes are the smallest and long oval the largest. Short ovals are a lot like a round, but much bulgier. The larger the eye, the less it rubs on the fibber you are using. Perle cotton needs an oval eye, as does crewel wool. Sewing thread does just fine in a round eye. Larger needles have larger eyes, but the basic shape does not change.
Diameters. This is what makes one needle a different size from another. The fatter the needle the smaller the size number. There are two different size ranges, one goes from 1-15 and the other from 13-28. In either range a big number means a small needle. As needles get smaller, they also get shorter. A lower number means a longer, fatter needle with a bigger eye.
Points. Tapestry needles are blunt, all the other needles have a sharp point. A glovers or leather needle has a triangular point with teeny cutting edges to cut a triangular hole in the leather as you use it. Some sailmaking needles have this, too. Even beading needles are usually sharp, but they are often so tiny that it's hard to tell.
Shape. Well, they are all long and skinny, but the eye creates a bulge or no bulge that will make a difference if you are doing bullion or french knots. For easier bullion knots, you want a smooth needle. A needle with a round eye has the least bulge. A needle with an oval eye has the biggest bulge. Rug needles and upholstery needles have curves in them, to do a 'scoop' stitch on fabric that you can't get to the back of.
Length. Some needles are supposed to be very long, like beading or milliners or doll making needles. Some are about as short as you would ever want to think about, like betweens that measure less than one inch. The length varies with the purpose, but the larger diameter needles are also longer than the same type of needle in a smaller size. So a size 18 tapestry is going to be longer and fatter than a size 24 tapestry.
So, lets put this all together and list what characteristics go with which type of needle.
Tapestry. Oval eye (smallest sizes have long oval), medium length, blunt. Sizes from 13-28. Common uses; cross stitch, needlepoint, counted thread work.
Embroidery/Crewel. Oval eye, medium length, sharp. Sizes from 1-13. Common uses; crewel work, ribbon embroidery, wool embroidery, smocking with speciality fibres.
Sharps. Round eye, medium length, sharp. Sizes from 1-13. Common uses; hand sewing, bullion knots or french knots in counted work, smocking.
Betweens. Round eye, short length, sharp. Sizes from 1-13, not often found larger than 7. Common uses; hand quilting, fine needlework such as shadow work embroidery or some French hand sewing.
Beading. Round eye, very long length, sharp. Sizes from 10-15 in the 13-28 size range. Common uses; beading, applying sequins.
Different manufacturers make needles a bit larger eyed, or fatter or longer or with different metals and finishes. This is just a general list of characteristics for some of the more usual types of needles.
(Thanks to Joan, the manager of G-Street Fabrics Notions department and to Barbara, manager of the Bernina department, for pointing me to the most useful articles.)