The term "wire wrapping" is used to describe many different styles of jewelry making which use wire (NOT beading/stringing wire). Some work is done with just wire, and some artists add gemstone or glass beads, cabochons (beads without holes), charms, or found objects or items from nature, such as shells. Some kinds of work are highly technical and technique-oriented, such as netting or weaving. Other kinds of wire wrapping are much more intuitive and organic where anything goes and your artists' eye will tell you when you are done. Either approach is fine.
Wire comes in different thicknesses, or gauges. The higher numbers are thinner wire, and are used for fine work. The lower numbers are thicker wire, and are used for frames or armatures, and for larger pieces of jewelry. Wire also comes in different hardnesses; most wire artists use dead soft (the most flexible) or half-hard (a little stiffer). To start getting the feel for working with wire, copper in gauges ranging from 16 to 28 and everything in between will give you a good range of gauges.
After you have a feel for it, you will likely want to graduate from copper to sterling or gold fill, or 14k gold wire. You might also want to try your hand working with SQUARE wire, which is used for many designs because several strands of wire will snuggle up and stay together (secured by a few wraps with round or half-round wire) due to the flat sides.
If you have never worked with wire, try some of the following to get a feel for it:
Put a 4 inch piece of 20 gauge wire between the jaws of your chain nose pliers, and bend it in half. Repeat with 26 gauge wire so you can feel the difference.
Next, put a 4 inch piece of 20 gauge wire between the jaws of your ROUND nose pliers, and bend the ends around to meet each other. Do this at the skinny end, in the middle, and at the fat end so you can see the difference in the sizes of the curves. Repeat with thinner and thicker wires so you can feel the difference in how they move.
Practice coiling a 24 inch length of 28 gauge wire onto a 6 inch straight length of 20 gauge wire.
Get a feel for how the different gauges of wire will behave when wrapping by using a gemstone or polymer clay donut. Practice wrapping the wire around the donut, and try to space your wraps evenly to train your eyes and hands to work together to get the wire to go where you want it to go.
Once you have played with the wire a bit and have a feel for it you will want to actually make something if you haven't done so yet. You are probably ready to explore some beginner to intermediate level tutorials, or to take a class should the opportunity arise.
Look for beginner-level tutorials that include "foundation skills". This includes making open loops, wrapped loops, coiling, and spiraling. Look for tutorials which will help you learn to use your fingers to guide and shape the wire, as well as those which require the use of pliers so you can practice using tools correctly to do the work.