There are 26 different types of nonprofits or not for profit organizations. In its most basic form, a nonprofit organization is one that, as its namesake states, does not claim a profit, and also serves the public interest. Generally, the purpose of the organization should be social, civic, charitable, educational, scientific, or religious. That covers a wide range of organizations!
Essentially, it means the organization utilizes all of the revenue available after normal operating expenses to serve its cause. Nonprofit organizations of all types have paid and volunteer staff, and generally employment taxes for them are no different from those imposed on their for-profit counterparts.
There are Trade organizations under IRS 501(C)(6), issue advocacy organizations such as Sierra Club or the NRA under IRS 501(C)(4), and so on – for an exhaustive list of types of nonprofits, see the chart below.
However, the vast majority of organizations that are charitable in nature and wish to be tax-exempt (including giving its donors the chance to claim charitable deductions) must be designated as a 501(C)(3) tax-exempt organization.