What laws govern trademarks?
There are a variety of laws that exist relating to trademarks, both on the state level as well as on the federal level. The most substantial trademark law, and generally the most relevant, is federal law. The Lanham Act provides detailed information regarding how one can obtain federal rights in a trademark, what those rights mean, etc. Specifically, the Lanham Act provides for a system of registration, whereby trademarks can be registered with the United States Patent and Trademark Office, which ends up granting the trademark owner obtains certain rights and benefits that generally exceed the trademark rights afforded by state law. Federal law also protects unregistered trademarks, but this protection is limited to the geographic area where the mark is actually used (whereas a federally registered mark is protected on a nationwide basis).
Many states have statutes similar to the Lanham Act, allowing for registration of a trademark on a state level. Such a state registration is not as strong as a federal registration, nor is it necessary if you obtain a federal registration. However, state registration is useful in those situations when you cannot obtain a federal registration.
Finally, all states have laws to protect unregistered trademarks. These laws do not come from state statutes, but are instead part of the common law. Common law trademarks are provided the weakest level of protection, however this protection is easy to obtain - you do not have to file any documents or seek any registration, you simply have to start using the mark.