It’s actually a very simple process, except that the Copyright Office has mashed its forms into one form and has not done anyone any favors in that revision.
You fill out the form at the linked page, you write a check for $45.00 to the REGISTER OF COPYRIGHT, and you send the form, the check and either one or two copies of the work to be registered (depending on whether the work is published or not) in to the US Copyright Office in Washington D.C. You then wait several months until a Certificate of Copyright is returned to you. Note that even though the registration certificate may take months to come back to you, the copyright is presumed to be registered when your mailing is received by the Copyright Office. You might therefore want to include some sort of proof of receipt with your mailing.
Copyright actually attaches the instant a work of authorship is affixed in a tangible medium, but the protection is less than ideal until that work is sent to the Copyright Office and registered. The federal courts, which have exclusive jurisdiction over copyright matters, won’t even look at a case where an unregistered copyright is infringed. No statutory damages are available to unregistered works, which means that the copyright owner must prove actual damages from the infringement to get any relief at all.
Since copyright is so easy to register, there really is no reason not to register it. Spend the $45 and get full protection for your valuable works of authorship.