A more experienced show person may come along and correct and improve this answer, but here's my understanding:
Each breed at a show that offers a brace class will award Best Brace to a brace of two dogs being shown together. At a group show like Montgomery (terriers only) Best Brace competition is the best brace from each breed competing against each other, the winner being Best Brace in Show.
Brace is a special class at some dog shows where two dogs are judged as one entry. The objective I guess is to demonstrate consistency of breed type by exhibiting two very similar dogs simultaneously. It's a challenging class with dogs as varied as Cairns can be. Other challenges include training the dogs to gait correctly and cooperatively so close together (note in the video Merril uses the tail to get one of her dogs back into position

), getting the dogs on and off the table without dropping one

, training the dogs to accept examination calmly while on the table together, etc.
Ask Liz - she, brave soul, entered a brace at her very first Montgomery. It was great to see.
Mostly it's just a fun class. It's typically offered at specialties and "group" shows (where all dogs are from the same group - terriers or hounds or sporting, etc.). There is another class, even rarer, called Team - where three or more (not sure about this) are exhibited in the same way. I've personally only seen dogs like Corgi's and Siberians and similar shown in teams. A team of Cairns would be a sight to behold, for sure!
Because these classes don't lead to championship points, they are called "nonregular" classes. Other nonregular classes you see at breed specialties which are fun are veterans (for older dogs), and Stud Dog and Brood Bitch. The latter are very interesting classes where a single dog (the stud dog or the brood bitch) is judged based on the quality of its progeny, who are in the ring with it. The younger dogs are examined, but the award goes to the parent.