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A trademark is anything that a company uses to identify the source of its products/services in its marketplace. Words, logos, sounds, numbers, and colors can all be trademarks.
The USPTO determines the right to register a trademark.
Courts determine the right to use a trademark.
Both registration and use turn on the issue of likelihood of confusion. In order for there to be a likelihood of confusion between two trademarks, both the (a) trademarks and (b) the goods/services covered by the trademarks must be close enough.
Registration provides significant advantages (such as recovering social networking usernames) but does not allow the registrant to prevent all uses of the trademark in all markets and in all situations. For example, trademark registrants cannot prevent a third party from registering a domain name. But trademark bullies may try stuff like this.