The accepted answer here is close, but not necessarily correct.
First of all, a common law trademark (not a Federal one) is established by use in commerce, not just business activity. The best example of this is a sale, but courts have accepted less.
Yes, there are requirements for a federal trademark applicant to search for names, but only in the principle register (a list of federal trademarks). There is no duty to search for an unregistered trademark, although if they did chose this mark out of bad faith it could have been brought up during the registration process.
If the trademark becomes registered, that registered trademark will be known as the "senior user" in commerce. This means that the registrant can expand use of the trademark in the entire United States, where as the junior user is limited geographically. A registered trademark also has the presumption of distinctiveness which is important in many proceedings.