Even if you have acquired the sheet music for a particular piece, this does not mean you are legally entitled to perform it. The license to perform copyrighted music publicly is conveyed separately from the purchase or rental of the sheet music itself. (If the music is no longer under copyright, you don't need to worry about this. You can perform it to your heart's content.) Performance licensing has to be obtained separately, for an additional fee, from ASCAP, BMI, or directly from the copyright holder. If you're renting music, the rental agreement will usually ask you to provide your ASCAP or BMI license number.
ASCAP and BMI are organizations devoted to facilitating performance licensing to make sure that the creators of music get paid the royalties they're due when that music gets performed or played in public. They have license categories for all types of businesses and organizations, from orchestras to, say, a restaurant that wants to have background music playing for ambiance.
Note: If you are a school orchestra or a church orchestra, your licensing requirements or fees might be different. The information described on this page is intended primarily for non-profit or for-profit orchestras that are not primarily educational or religious organizations. Those organizations either require different categories of licenses or are exempt entirely.
An orchestra can purchase an annual blanket license for a flat fee that allows you to perform any of (and as many of) the works in their catalog. I haven't worked with BMI directly, so I'm not sure how they do things. For ASCAP, the license type your orchestra will need is called "orchestra" (not "concert"). They have a tiered pricing structure depending on your organization's annual budget. The annual blanket license for the lowest tier, for orchestras up to $250,000 per year or something like that, is about $340. This blanket license is the only thing you can get. You can't license an individual piece of music for a specific date. Even if you're only playing one single piece in one concert, for one day of the entire year, there's no way for your orchestra to get anything other than the standard annual blanket license (trust me, we tried). And, no matter when you purchase the license, they're always up for renewal in September, and they are not pro-rated.
Also, once you have purchased an ASCAP license once, it seems very difficult to get rid of it later. They send you an annual renewal form with no option for "do not renew". You have to go to a lot of effort to convince them of why you don't need to renew your license because you're not planning to perform any pieces in their catalog, and they will pull up your website and try to find your performance repertoire. Also, they don't respond quickly (or at all) to e-mails or inquiries made through their web form, and they don't always call you back if you don't get through to the right person on the phone when you call.
Technically speaking, an all-volunteer community orchestra might not actually need performance licensing. As detailed in Section 110, Clause (4) of the US Copyright Code, it is not a violation of copyright to perform a copyrighted piece of music in public if there is no admission being charged, and none of the performers or organizers are being paid. However, good luck using this exemption in practice. The RCO tried, and it didn't work. The music publisher we were trying to rent from wouldn't accept this argument and said the only thing they would take was an ASCAP license number or a letter from ASCAP on our behalf saying we didn't need one. We talked to ASCAP, and they refused to provide us with such a letter. We wrote to the copyright holder (a foundation for a deceased composer), and they just referred us back to the publisher. So, we had to purchase the ASCAP license anyway even though we still think we didn't legally need it.
I have been told that concert bands can get licensing through the Association of Concert Bands for a lower price plus a membership fee, so if you're starting a band, definitely check that out.
Disclaimer: I am not trained in law. I've done my best to understand and explain a complex subject, but I can't be held liable for any trouble you might get into following this advice (though by all means, let me know if I've given bad advice!). If you have serious legal questions, please consult a lawyer specializing in music copyright law. I had an extremely helpful phone consultation with Mr. Greg Victoroff in Los Angeles, who himself plays in a community orchestra.