Trademarks do what their names imply—they mark or brand something, usually a commercial item.
A graphic symbol, like the Nike swoosh is a trademark. Because the Nike swoosh is a registered trademark and is used to distinguish Nike products from other manufacturer's products, it may only be used by Nike. Trademarks can also be words, like slogans such as Pepsi's "the Joy of Pepsi™."
Registering your trademark with the US Patent and Trademark Office will help if you need to prove anything legally, but it's not required.