This is general, factual information, not legal or tax advice — for your specific situation, a nonprofit attorney or accountant should always have the final word. That said, here's the basic shape of how this works in the United States.
Churches are automatically considered tax-exempt under section 501(c)(3) of the tax code in many cases, without needing to file for formal recognition — but many still choose to apply for a formal determination letter from the IRS, since it provides written proof of status that donors, grantors, and banks often want to see.
Tax-exempt status affects more than income tax. It typically also enables donors to deduct their contributions, and often qualifies the organization for reduced postal rates and other nonprofit-specific benefits.
None of this replaces professional guidance — the requirements around church-specific exemptions, state-level registration for charitable solicitation, and ongoing compliance vary by situation, and getting it wrong early is far more expensive to fix later than getting proper advice up front.