Do I have a right to my name?

I registered my domain name in Feb 2011. I have been using my domain name as my business name. I just checked trademarks, and there is a company that registered the name in March of 2011. It's not the same exact name, it is ONE letter of difference. Like "Apple" and "Apples".

Does the fact that I was using the name before them count for anything? I would think that since they registered it, I am going to have to change my name. Any help?

There was a case of etoys.com suing etoy.com where etoys was bigger but etoy had their domain name first - the suit was eventually dropped. In general it's a bit messy but I don't think it will be clearly settled until a judge rules on it.

If your domain name is "apples" given that Apple has a history of taking domain names like 'applesucks.com' - which clearly should not cause confusion in the minds of consumers - then I'd say you're in trouble unless you have the money to defend yourself in court. In general, the deeper the pockets of the 'other guy' the quicker you might want to stop building any kind of brand with that name.

All that said, I personal would side with you, were I on the jury and if it were a jury trial.

Another thing to consider is selling your domain to them, if that would work for you.