FMCSA USDOT

Companies that operate commercial vehicles transporting passengers or hauling cargo in interstate commerce must be registered with the FMCSA and must have a USDOT Number. Also, commercial intrastate hazardous materials carriers who haul quantities requiring a safety permit must register for a USDOT Number.
The USDOT Number serves as a unique identifier when collecting and monitoring a company's safety information acquired during audits, compliance reviews, crash investigations, and inspections.
See: http://www.fmcsa.dot.gov/registration-licensing/registration-USDOT.htm

FMCSA Operating Authority

FMCSA operating authority is also referred to as an "MC," "FF," or "MX" number, depending on the type of authority that is granted. Unlike the USDOT Number application process, a company may need to obtain multiple operating authorities to support its planned business operations. Operating Authority dictates the type of operation a company may run and the cargo it may carry.
See: http://www.fmcsa.dot.gov/registration-licensing/registration-OP.htm

Application of USDOT or Operational Authority data

FMCSA assigns a USDOT for carriers who move freight between states (Intrastate) as well as one or more Operational Authorities which designate what type of freight they are classified/authorized/registered to transport; Each operational Authority can be resolved to a single Carrier.

• FMCSA requires the Carrier to use their Name, Address, and potentially their USDOT (pending proposed rules) on an HOS Log.
• States require the Operational Authority to be provided for Fuel Tax purposes

This diagram Illustrates this situation:

How can I find a companies USDOT or Operating Authority numbers?

Direct Link to Company Snapshot: http://safer.fmcsa.dot.gov/CompanySnapshot.aspx

FMCSA SAFER main page: http://safer.fmcsa.dot.gov/