Cycle mitts provide padding and protection to two of your main contact points with the bike, rather like shorts with a chamois do for another contact point. Like padded shorts they're not vital for utility cycling but they vastly improve comfort if you're riding fast, far or on rough ground. The less shock absorption you've got from tyres, bar tape, or even forks, the more you'll need them.
Mitts also provide palm protection from cuts and grazes in the event of a fall. Their main purpose, however, is to prevent your hands from aching, tingling, or going numb. Human hands aren't designed to be weight bearing, especially hands that are already more prone to aches and pains from hours of clicking mouse and keyboard. Next to shorts and, yes, ahead of helmets, they're probably the most important bit of cycle wear you can buy.
The area that most needs padding is the heel of the hand opposite the thumb, under which run the ulnar and median nerves. Pressure on these can lead to numb or tingling hands. Padding at the base of the thumb and fingers can also improve comfort but is less important. Gel padding is excellent, because it tends not to compress and 'bottom out', but dense foam rubber or neoprene also work.
Mitts can become rank with sweat and snot after a few rides. While the towelling thumb is handy on th move, it can also be a breeding ground for bacteria (which you'll then wipe around your face.) Wash mitts before they get too wiffy and don't store mitts for too long.
Finger loops or tabs help in removing close-fitting mitts, especially those lacking a Velcro strap. Reflective piping is useful for summer nights (in cold weather you'll be in gloves). Protection for the back of the hand prevents scrapes when mountain biking but isn't vital on the road.
Design and fit
Mitts use elastic and/or a Velcro strap to help them fit snugly and securely n of this small patch of (The lifespan of of this small patch of Velcro can be short). To prolong wear-life and improve comfort, many are reinforced at the thumb arch and across the base of the fingers. Stitching is often a weak point on cheaper synthetic mitts, which may not last a season.
Traditional cycle mitts had a tough, padded leather palm with a string back for ventilation. You can still get them, and even modern mitts usually have a leather or synthetic-leather palm. Uppers are now usually lightweight, stretchy polyester. Towelling or fleece on the thumb is useful for wiping sweat - or snot.