Rivets are small metal studs hammered through two or more layers of denim at points that come under repeated strain, most visibly at the corners of the front pockets, and sometimes at the base of the fly or other high stress seams. They work by clamping the layers of fabric together so the stitching at that point cannot pull apart under load.
The rivet is one of the oldest and most recognisable features of jeans, dating back to the original 1873 patent held by Jacob Davis and Levi Strauss, which used copper rivets to solve the problem of pocket seams splitting under the weight of tools and contents in workwear. Modern rivets are sometimes replaced with reinforced bar tack stitching instead, particularly on styles designed to avoid marking furniture or car seats, since exposed metal rivets can scratch surfaces.
Checking that rivets sit flat and are not loose is a simple way to judge the construction quality of a pair of jeans before buying. See our jeans history guide for the full story, and our buying guide for more on assessing construction.