Quickdraws are made up of two carabiners joined together by high-strength webbing. When climbing, one carabiner is clipped to a point of protection, for instance a bolt hanger or camming device. This carabiner is usually able to move freely within the sewn sling to reduce the 'pull' on the protection. The other carabiner is clipped to the rope. This end is often secured within the sling by a tight rubber 'tadpole' which prevents the carabiner from twisting, reducing the possibility of cross-loading and keeping it well positioned for clipping the rope. The sling increases the distance between protection and rope (hence why they're often called 'extenders'). This allows the rope to run more freely with less risk of dislodging the protection.
Traditional climbing with quickdraws
For traditional climbs, quickdraws with lightweight wire gate carabiners are the norm. The wire gate is especially useful on winter climbs as it won't freeze shut. Thinner, lighter slings also help cut the weight of your rack on the harness. Carrying a variety of lengths from 10-12cm (short) through 15-18cm (medium) and up to 25cm (long) allows you to manage the line of the rope on meandering trad routes and when crossing roofs. A rack of a dozen quickdraws will usually suffice for most pitches.
Sport climbing with quickdraws
For sport climbs, quickdraws with sturdy solid gate carabiners are the norm. Chunky carabiners put up with constant abuse and regular falls. Wider slings are much easier to hold when working routes and hauling back to your high point. A straight gate makes bolt clipping easy while a bent gate facilitates rope clipping. 10-12cm lengths are fine for most straight-up routes, but it's also worth carrying a few longer sport draws for occasional use. A rack of a dozen quickdraws will suffice for most pitches but long European sports routes might require far more. Guidebooks often give the number of bolts for any route so you can check you have enough before setting off.