The first step is to cut a piece of wire about 11 inches long. Then bend your wire about 1 1/2 inches down, and thread a bead on it.
Next, twist your wire around, more than half way up. (Or, like the photo, you can string beads up for the tail.)
Then, string a bead on going through both wires. This will be your largest bead, because it represents the body of the dragonfly. After you have your bead on, make a loop with your long piece of wire, like in the photo. (I don't use any tools to do this, I just eyeball it and shape it with my fingers.)
Now, you're going to make three more loops, crossing over the middle as you go. It will look like two figure 8's on top of eachother.
Next, take your longer wire and wrap around your two left loops, cross over the middle and wrap around your two right loops. You're just securing them in place with your wire, by wrapping around them. You should have something that looks like this when you're done:
Then all you need to do is thread on a smaller bead (this will be your head) and curve your wires around to make little antenna. (You'll want to snip your wires to the same length before you do this.) If you have a wire bending tool, you can use that, otherwise I sometimes use a small paintbrush handle or other such thing to wrap the wire around. Also, fan your wings apart with your fingers.