A bridle was a harness for horseback riding.
One component of the bridle was the bit, which was placed in their mouth and used by the rider to communicate with the animal, to "champ" or "chomp on the bit" was an Earth idiom used to mean to be impatient or frustrated with a delay.
When Captain Jonathan Archer denied Trip Tucker's request to join the boarding party of a derelict Axanar cargo ship in 2151, Tucker complained to the captain, that "[y]ou're chomping at the bit to find out what's in there. Why should I be any different?" (ENT: "Fight or Flight")
In 2254, illusory versions of two horses owned by Christopher Pike – Tango and Mary Lou – each wore a bridle. The bridles and everything else in the illusion were telepathically created by the Talosians. (TOS: "The Cage", "The Menagerie, Part I", "The Menagerie, Part II")