First, go to your profile page. (When logged in, either click on your avatar image or the "Profile" link under My Account.)
Now, under the Profile tab, click "Edit." This will let you change your Display Name and/or your Forum Signature. Make the desired changes, then click "Save Changes." That's it! You now have a fancy shmancy forum sig.
Now it will seem a little misleading, because the page will say you can use BBcode, but HTML will be stripped. Well, that's true - but if you use BBcode in your sig, after you save it gets converted into HTML (and it shows the HTML code on your profile page, however, in the forum it displays correctly.)
Important: At this stage, if you go in and alter the HTML code in your signature, it will get stripped! (Again, I know because I tried.) So don't do it! If you're happy with how your signature looks in the forum, then just leave it alone...