There's no "common law copyright" in the US. There's common law trademark, but copyrights and trademarks are different things. All copyright is based on federal copyright statutes, and an unregistered copyright is just called an unregistered copyright. What an unregistered copyright gives you is almost the same thing as a registered copyright: the exclusive right to exercise various rights (reproduction, distribution, public performance/display, etc.) Registration is required before you can actually file an infringement lawsuit, but it's OK if you only registered it after learning about the infringement.
The main reason to register the work early is that you might be able to collect higher damages if it was registered before the infringement. If the work was unregistered when the infringement happened, you can only collect actual damages. If it was already registered, you can instead opt to collect statutory damages. Statutory damages don't require you to show actual loss; instead, the court awards between $750 and $30,000 per work based on what it considers fair. The lower limit can go down to $200 if the infringer had no reason to think they were infringing, and the upper limit can go up to $150,000 if the infringement was willful. You can also collect attorney's fees if it was already registered.
In the comments, the question was raised of whether someone else might be able to register the copyright for your work before you and what would happen then. When I say "registration is required before filing a lawsuit," the requirement is that you submit the registration paperwork properly, pay the fee, and wait for the Copyright Office to act on the application. If they accept the registration, you're all set. Even if they refuse, though, you can still file an infringement claim -- you just have to also serve the head of the Copyright Office, who has the option of jumping into the case to argue about whether the work could legally be registered. You can also sue the Copyright Office to get judicial review of a denial under the Administrative Procedure Act (refusal still lets you file infringement lawsuits, but you can't get statutory damages or attorney's fees against an infringer unless the Copyright Office actually accepted the registration).