Neither a copyright nor a trademark is required to use that name on a website or otherwise in commerce. In fact, a business name is not copyrightable and because your name is primarily descriptive, it is unlikely that a trademark would be granted. However, most states have laws against unfair and deceptive business practices that could provide some protection for your brand.
No. You have it backwards. Filing a copyright or trademark registration does not convey to you the right to use a work or a mark. It's the opposite. By actually using a trade name or by reducing your idea to a tangible expression, you THEN acquire the right to file a copyright registration or trademark application.
What you have to worry about, and investigate, is whether someone else is already using that name. You should first do a quick search of the trademark records (www.uspto.gov) for the name you would like to use to make sure someone else is not already using it and has filed their own registration application.
You don't need to copyright or trademark your name to use it, since you have automatic copyright and trademark protection available (you should use appropriate notations - the "c" in a circle or a small "TM" next to the mark or material in public uses).
HOWEVER - this strikes me more as a fictitious name question. The name "Penn Hills Tai Chi Society" is a fictitious name and PA law requires you to register fictitious names with the Pennsylvania Department of State Corporation Bureau (see http://bit.ly/cfXkUc for information on how to register). Registration is simple and costs $75. I'd urge you to take that action ASAP.
Of course, as with all of my online answers, my advice is limited by the brevity of your question and the facts provided. Additional information would be required to provide definitive legal advice, so this answer isn't intended to, and does not, create an attorney-client relationship.
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Forget copyright law as a source of protection for "Penn Hills Tai Chi Society" because names such as that are not eligible for copyright protection. As for trademark protection, the name is not registrable as a trademark because it is both geographically descriptive and is simply the generic name a group involved in offering Tai Chi services.
There are four things you can do however:
(1) as Attorney Tamanini suggests you should register "Penn Hills Tai Chi Society" as the fictitious business name of your unincorporated business association [i.e., your "Society"],
(2) you should affix the "TM" symbol to the upper right hand corner of the word Society whenever the name of your group is displayed in a title or otherwise prominently,
(3) you should consider creating a logo trademark for your group and, if it's consistent with your creative senses, weave the name into the logo somehow or have it otherwise very close by,
(4) once you create a logo trademark you should chat with your own trademark attorney to see if your logo can be registered and, if so, how best to accomplish that [state registration? federal registration? on what goods or services? what to disclaim if anything? etc.].
The above is general information ONLY and is not legal advice, does not form an attorney-client relationship, and should NOT be relied upon to take or refrain from taking any action. I am not your attorney. You should seek the advice of competent counsel before taking any action related to your inquiry.