The main thing you're concerned with is whether the organization is allocating enough of its income to actual services rather than spending money on overhead, fund- raising, events, or savings for the future. You're going to be most interested in the percentage of funds that goes to administrative costs, since those are the dollars not going to the people whom the charity is intended to benefit.
Basically, a group should be spending no more than 25 percent of its income on overhead, with at least 75 percent going to actual services. Anything more than 75 percent is really, really good. Anything between 50 and 75 percent should make you wary and want to ask some questions before donating, e.g., why is so much money going to expenses other than the work of the charity?
Most nonprofits list this information in their annual report, which you can find on their website. You can also ask any group to send you this information by mail or e-mail. Smaller groups may not produce a formal report and may not even have a website. Don't hold that against them, though, since they, too, should be able to send you financial information by mail or e-mail if you request it.