FIPS standards drive the core requirement or approval statement. FIPS 140-3 defines cryptographic module requirements. FIPS 197 defines AES. FIPS 180-4 and FIPS 202 define secure-hash families. FIPS 186-5, FIPS 203, FIPS 204, and FIPS 205 define signature and PQC standards.
The NIST SP series often tells you how to apply those standards in practice. In the FIPS 140-3 ecosystem, the SP 800-140 series modifies annex requirements for CMVP use. In the algorithm ecosystem, publications such as SP 800-175B, SP 800-89, SP 800-208, SP 800-227, and SP 800-185 fill in application, assurance, or operational detail.