A mechanic is a discrete system within a game’s rules that governs how a specific kind of action or interaction works. Drawing cards is a mechanic. Rolling dice to resolve uncertainty is a mechanic. Bidding, auctioning, drafting, tile placement, and resource trading are all mechanics.
The value of thinking in terms of mechanics is that it lets you compare games across traditions. Poker and bridge are different games, but they share mechanics (hand management, hidden information, betting or bidding). A tabletop role-playing game and a board game may both use dice resolution but differ in whether the dice decide success or failure, or degree of outcome. Naming the mechanic makes the comparison precise.
Mechanics are the layer between rules and experience. Rules define what is legal; mechanics define what it feels like to play. Two games with the same win condition but different mechanics (auction vs. dice roll) produce different player experiences, even if the strategic depth is comparable. Game design is the art of choosing and combining mechanics to produce a desired experience.