First and foremost, always make a backup before you attempt to upgrade!
Log on to the vBulletin Member's Area and download the new version. Unzip the files, retaining the directory structure.
You should see a folder named upload and another folder named do_not_upload. Rename the upload folder to whatever name you gave your vBulletin directory on the server.
If you've made any changes to the php files of your current vBulletin installation, compare the files and see if Jelsoft has made any changes to the new version. If not, delete the file on you hard drive so you don't overwrite your changes. If they have changed the file, you'll have to reapply the modifications. (I prefer to do this after I'm sure the upgrade was successful though, so if there's a compatibility problem you know it's the mod and not vBulletin itself.)
If you've previously changed the names of your admincp and modcp subdirectories on your server, change the subdirectories on your local files as well before uploading.
Log on to vBulletin and turn it off under vBulletin Options, Turn Your vBulletin On and Off.
Some people advocate deleting your entire vBulletin directory at this point and uploading the entire new version as if it was a new install, but you would lose any extra php or xml files you've added. Only do this if you truly want a fresh start.
Upload the files to your server, overwriting everything.
Point your web browser to
http://www.yoursite.com/forum/install/upgrade.php (replacing yoursite.com/forum with the actual path to your vBulletin).
Enter your customer number, the one you use to log on to the vBulletin.com web site.
Generally all you have to do now is keep clicking the Next Step button and watching for errors. If you spot any, copy the error message and run a search on it, or ask for help it you still can't figure it out.
Once it says you're done, delete install.php at the very least... better yet, delete the entire install directory.
Log on to your admincp and see if you have to revert any templates. Visit your forum and make sure everything is working.
Reapply any changes to the vBulletin files you made before upgrading.
Finally, turn the forum back on