Approach shoes are a hybrid of hiking shoe and climbing shoe — built for the hike to the crag, but capable of easy scrambling and even some technical climbing. They have a stiff midsole for hiking comfort, a sticky rubber outsole (often the same compound as climbing shoes), a "climbing zone" of flat smooth rubber at the toe for edging on rock, and a low-profile design closer to a runner than a boot.
The best-known examples are the La Sportiva TX2 and TX4, the Scarpa Crux and Mescalito, and the Five Ten Guide Tennie — names familiar at any popular trad or alpine crag. They're indispensable for routes with long approaches, multi-pitch climbing where you carry shoes for the descent, and any time the path to the climb involves slabby rock or class-3 scrambling.
For purely indoor or single-pitch sport climbing, approach shoes are overkill — running shoes or hiking shoes do the job. They earn their keep when the approach itself is technical, when you're clipping them to your harness for the walk-off, or when you want one piece of footwear that handles the day from car to summit.