Quick plug with the long answer...
Quick Formations - Trademark Registration, Registering A Trademark
Now the short answer;
You would register a trademark usually through a Trademark Attorney, I'd recommend using one of them because they would use their expertise to advise you on the bets format to apply for your registration and be able to process it in an efficient manner.
You can do it directly with the Patent Office for a UK registration, but you would need to consider the industry sectors, trademark format and style yourself.
You can also do searches through the Patent Office website for existing trademarks that may be in existance.
If you register a trademark of a logo that someone else might already be using, then you may supercede their claim to that image if you are in the same industry sector (see comment re industry sectors above), but they could also object against your application and override it. Before a trademark application is approved it goes into a pending state for people to object before it is given final approval.
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Trademarks
