Push-button (or pin lock) ski poles use a spring-loaded button that clicks into preset holes along the shaft, locking the pole at specific length increments. They're fast to deploy—just pull the section out until the button snaps into place—and very reliable once fully engaged, since there's no clamp to forget to tighten or twist mechanism that can slip.
These are most common on folding poles designed for backcountry touring or travel, where compact packability is a priority. If you want a pole that collapses down small for stashing in a pack, push-button is often the mechanism of choice.
The main trade-off is that you're limited to preset lengths rather than infinite adjustability. If your ideal length falls between two holes, you'll have to pick one or the other. For most skiers the increments are close enough, but if you're particular about exact pole length, an external clamp style may be a better fit.