Within the field of accounting, the process of maintaining financial records and auditing financial records for purposes other than tax collection is called "financial accounting" as distinct from "tax accounting" and as distinct from "managerial accounting" which is the internal use of financial information for the purposes of managing the entity and improving its performance rather than for outside oversight of an entity.
Financial audits (as opposed to "performance audits") almost always pertain to either financial accounting or tax accounting. Obviously, tax audits are done by tax collection officials.
In the United States, financial accounting standards for non-profits are primarily set by a private organization controlled by the accounting profession called the Not-for-Profit Advisory Committee which provides advise that is usually followed to the accounting profession controlled private Financial Accounting Standards Board that defines "Generally Accepted Accounting Principles" (GAAP) in the United States for the for profit and non-profit sectors. This is the main body that sets industry accounting standards particular to non-profits and for fund accounting.
The private sector financial standards set by organizations like the FASC and NAC which are controlled by the accounting profession, are then adopted by reference by public or private organizations that want to specify accounting rules without having to devise them from scratch themselves.
These financial accounting standards are subject to specific tax law rules set by the IRS (under the Internal Revenue Code, ERISA, and related regulations and regulatory guidance) and state tax collection agencies, and by other industry specific regulations.
In cases where non-profits issue publicly held debt, the Securities and Exchange Commission and/or other applicable state and federal securities regulators, and the governing body of any exchange upon which the bonds issued by the non-profit are traded can also impose accounting standards.
State probate statute also set some of the accounting standards for non-profits such as the rules for distinguishing between principal and interest.
Public or private donors to non-profits usually make grants that condition receipt of grant funds on a right to audit the non-profit, and investors in bonds issued by a non-profit generally have either a contractual or statutory right to obtain audited financial statements from the non-profit as well. These audits would usually be performance by Certified Public Accountants in private practice (and regulated under state administered licensing schemes) who are hired on an audit by audit basis.
In addition to tax collection authorities, and private people who have audit rights because they have invested in bonds from the entity or have given grants to the entity, state attorneys-general will generally have broad supervisory authority over non-profits including the same kinds of rights with respect to a non-profit that a beneficiary of a private trust would have (although sometimes a different official in state government is given this authority).
State attorneys-general are given this authority so that there is someone with jurisdiction to oversee the affairs of, and prosecute misconduct in, charitable organizations with no well defined individuals or entities that are permanent beneficiaries of the charities.