Nonprofit organizations must file an annual Form 990 with the IRS. The form summarizes finances, lists board members with salaries of top officers, and provides other general information. Nonprofits are required to share their 990 forms by posting on the web or providing information during business hours. Most are now becoming available on the web at Guidestar.org, at the web sites of individual organizations, and at other association sites.
Depending on the type of nonprofit (or "not for profit"), they may also file an annual report with the Secretary of State's Office, Department of Corporations. You can request these reports directly from the State when they are not online. However, some states make annual reports available online.
Another interesting fact is that annual reports from an organization will usually provide information and details not in the official federal and state filings, whereas, the official filings may provide details not in annual reports. What to use could depend on what you want to know.
Various laws regulate where and when reports must be filed, what they must contain, and how the reports must be distributed, and this in turn depends on the type of nonprofit in question. The IRS website has further detailed explanations about types of nonprofits.