When you talk about a non-profit business, you are generally talking about two things: being registered as a not-for-profit company in a state, and being recognized by the IRS as a tax-exempt 501(c) — typically 501(c)(3) — organization. Without approval by the IRS, your business will pay income tax whether or not you are a non-profit.
Limited liability companies are very flexible business entities, and this applies to ownership restrictions as well. The owners of an LLC are called members. A single member can own an LLC by him or herself, but a partnership of members, a group of members, or even a group of businesses can own an LLC. Limited liability companies are creatures of state law, which means every jurisdiction in the United States will have slightly different rules. The limitations on ownership, however, are fairly consistant across the country.