Trademarks show the ownership of a name, such as a brand or a product name. Registering a trademark shows ownership of the mark and any design made part of the trademark.
Registering either a patent or a trademark makes public the ownership of the invention or name and gives the owner legal rights to prevent other people from using the patented invention or trademarked name.
In the United States, patents and trademarks are usually registered through the US Patent & Trademark Office.
To protect trademarks internationally, you can register the trademark in other countries as well.
Trademark and patent attorneys specialize in this area and can advise you about the process and costs involved.
It is also possible to handle the registration yourself, but the process is complicated. More information is available on the USPTO site.