When wire is referred to by its gauge, it is a reference to the diameter of the wire. There are two standards of gauges generally used in jewelry to state the size of wire. American Wire Gauge (AWG, also known as Brown and Sharpe), and Standard Wire Gauge (SWG, also known as Imperial or British Wire gauge). The smaller the diameter of the wire, the bigger gauge number. For instance, a 20 gauge (AWG) wire is 0.0320 inches or 0.812 mm, and 12 gauge (AWG) is 0.0808 inches or 2.053 mm. AWG is slightly larger in size than SWG. 20 gauge in AWG is 0.0320 inches and 20 gauge in SWG is 0.036 inches.
In addition to various sizes, wire is also available in different shapes, from full round, half-round, to square. Full round wire is just as it sounds, a fully round, cylindrical shaped wire. Half-round wire is flat on one side and round on the other. And square wire has four equal sides throughout the length of the wire. Using different shapes of wire can give your pieces unique dimensions. An example of this is when using half-round wire for wire wrapping, as it can lay flat against a straight surface and appear round to the viewer.
Wire is also measured by its hardness. For jewelry, hardness is often described by three terms: dead soft, half-hard, and hard. Dead soft wire is extremely soft and pliable. This wire is great to use for wire-sculpted jewelry and for making rounded shapes and spirals; however, dead soft wire tends to lose its shape if not handled properly. Half-hard wire is slightly stiffer than dead soft wire. This wire is good for making tight angular bends, and tends to hold its shape. It does not do well with making round shapes or spirals. Hard wire is very stiff and tends to spring back after being bent. When enough force is applied, hard wire makes crisp sharp corners, and while it is generally much harder to work with, its shapes tend to be more permanent than half-hard wire.
Any time you manipulate wire, you change it's molecular structure, and this is known as work hardening. The more you work with the wire, the harder it becomes. As a wire becomes harder, the more brittle it becomes. You can experience this yourself by bending a wire paper clip back and forth until it becomes brittle and breaks. The only way to make wire soft again is by heating it with a torch or in a kiln.
I hope this short post has helped explain some of the different aspects of wire and some of the terms used in its description.