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A letter of concurrence is authorized under 90.20(c)(2) (Public Safety Channel Pool) and 90.35(b) (Business Channel Pool).  It's not explicitly stated in the Public Safety section of the rules, but is in the Business Pool section:

 90.35(b)

(iii) For frequencies above 150 MHz, applications for new or modified facilities on frequencies shared prior to radio service consolidation by the former Manufacturers Radio Service, the Forest Products Radio Service, the Power Radio Service, the Petroleum Radio Service, the Motor Carrier Radio Service, the Railroad Radio Service, the Telephone Maintenance Radio Service and the Automobile Emergency Radio Service may be coordinated by any certified Industrial/Business Pool coordinator. However, in the event that the interference contour of a proposed station would overlap the service contour of an existing station licensed on one of these previously shared frequencies, the written concurrence of the coordinator associated with the industry for which the existing station license was issued, or the written concurrence of the licensee of the existing station, shall be obtained. For the purposes of this § 90.35, the service contour for UHF stations is the 39 dBu contour; and the interference contour for UHF stations is the 21 dBu contour; the service contour for VHF stations is the 37 dBu contour; and the interference contour for VHF stations is the 19 dBu contour.
 

Translation:  If a frequency coordinator encroaches on any of the special sub-pools of channels within the general pool of all channels, then they need to get permission from either that special pool's designated coordinator, or the actual licensee that would be interfered with if the new application were to be granted.

This permission is called a letter of concurrence.

Letters of concurrence are also often required by coordinators directly from a new applicant in almost any case in which they (the coordinator) have the slightest doubt about any frequency that a new applicant may be asking for.  It's an effective method to deflect any future complaints about a license granted, and also an effective way to stop an application in it's tracks, because many license holders - especially government agencies - either don't understand what is being asked, or will just automatically give a "no" answer to any requests for letters of concurrence.

Sometimes they are right to do so, sometimes they are not.  If a new applicant is convinced (and can prove) that their proposed system will not cause harmful interference to  existing stations, then they may be able to convince the coordinator to accept interference studies in lieu of a letter of concurrence.  More likely, however, the new applicant must either change frequencies, or request that the application - along with the interference studies - be submitted to the FCC over the objections of the coordinator, and allow the FCC to judge the merits of the case.

Sometimes a coordinator will request a letter of non-interference instead of, or along with a letter of concurrence.  Generally, this is when the coordinator really doesn't believe that there will be any interference, but wants to be able to protect themselves from any future complaint.

Letters of non-interference are nothing more than a re-statement of a potential licensee's responsibility to be a "good frequency neighbor" and follow the rules for the shared use of channels.

There are draft examples of a letter of concurrence and a letter of non-interference in our library.

Here are some examples of letters of concurrences that have been used in the coordination process:

APCO Coordinator Letter of Concurrence.

Kentucky State Police Letter of Concurrence.

Knox County Letter of Concurrence.

Clay County Letter of Concurrence.

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