Three families of grippers, three very different tradeoffs
Everything a robot arm does in the world funnels through whatever tool is bolted to its wrist, and for manipulation that tool is almost always some kind of gripper. Parallel-jaw grippers use two fingers that close together in a straight line, and they're the default choice for rigid, regularly shaped objects like boxes, bottles, and mechanical parts. They can apply real clamping force, which makes them good for objects that need to be held firmly during a fast motion, but that same rigidity means a poorly placed grip on an oddly shaped object can miss entirely or crush something fragile.
Suction grippers use a vacuum cup, or an array of them, to grab an object by sealing against a flat surface and pulling air out. They're extremely fast to engage, which is why they dominate in warehouse picking of boxes and packaged goods, and they work well on flat or slightly curved surfaces like cardboard, plastic packaging, or produce. Their weakness is exactly the flip side of that strength: anything porous, textured, wet, or without a flat face to seal against defeats them, and a small air leak can mean a dropped object mid-transit. Soft or compliant grippers, built from flexible materials or underactuated linkages, deform around whatever they touch instead of requiring a precise geometric fit. They're the best option for irregular or delicate items like fruit or oddly shaped tools, but they're generally slower to actuate and provide less raw holding force than a rigid parallel jaw.