I haven't cycled for many years already and long ago when I was cycling I didn't pay much attention to equipment. Now I see that many cyclists wear a certain kind of gloves - that only cover the palm and leave the fingers uncovered.
What's the purpose of wearing such gloves (and not usual gloves for example)? In what non-cycling activities can such gloves be useful?
Cycling gloves do a few basic things:
Of course, if none of these are problems for you, it's fine to ride without them. Other types of gloves should offer similar benefits too (wool gloves with "grippy" palms are nice in cold, wet weather). I try to ride without cycling gloves for rides up to an hour to toughen my palms and reduce the bizarre tan lines the gloves create. But for longer rides (especially randonneuring, which is 200km+) I can't ride without them.
What else can you use them for? Basically anything where a padded palm would help. They make great workout gloves, and I've occasionally used a pair for paddling, doing yard work or helping someone move.
Cycling gloves come in different varieties (like full-fingered or half-fingered). Typically:
Cycling gloves dampen the vibrations coming into your hands from your handlebars.
I find that if I cycle for more than one or two hours without gloves I slowly loose feeling in the tips of my fingers and I get a tingling feeling in the pinky side of my palms near my wrist. This slowly spreads to essentially make both my hands feel numb. It goes away quickly when I get off the bike at first, but during cycling holidays where riding a bike is a daily activity I ended up with numb hands essentially all the time. Simple cycling gloves have solved this for me.
Note that front suspension would probably address at least part of the same problem.
For bmx, the greatest benefit is grip. If your hands are sweaty, you'll have to apply a lot of extra force to prevent throttle grip. Gloves eliminate all that. Also, the padded palms work nice to prevent meat paw, terry cloth thumb is good for wiping sweat away, and there is usually some tacky material on the fingers for good brake lever grip.
Gloves do help with hand numbness/discomfort, but the primary benefit to me is safety. If you fall and take a handlebar spill, your hands won't be torn up if you instinctively put your hands out to protect you.
Padding and warmth are covered in other answers, but in the winter I have a third use. In the warmer weather my fingers swell slightly, which doesn't happen so much in the winter, so my wedding ring has a tendency to be looser in colder weather.
Long fingered gloves stop me from worrying so much about it slipping off when on bumpier rides.
When you fall off a bike your natural reactions will lead to putting your hands out to stop your fall. Your hands are quite likely to hit the ground first so gloves provide some protection against that.
Comfort is another factor - damping vibration and providing protection from the elements.
Depending upon your climate you may want different gloves for different seasons. I wear motorcycle gloves on my bicycle in the winter.
To answer the second question. Gloves like that are great for working on cars... The padded palm keeps you from tearing up your hands on old rusty parts. The top of the hand is covered for when you slip and bang into sharp rusty parts. The fingers are exposed so you can still easily grip small parts. I find they work much better than traditional "full finger" mechanics gloves...
aside from the warmth and padding, I also wear them to protect my hands in case of wipeout. grinding gravel into my palms doesn't sound like a fun thing to do.
One thing that I use my gloves for on nearly every ride is putting my palm down on the tops of my tires after going through a section of stone dust or cinders. In fact, just today I was forced to ride through a section where some fresh blacktop spillage stuck to my tires. A quick drag on each wheel and the tires were completely clean again.
One other purpose for gloves is that if you have rubber grips, for example grip shifters on a hybrid or mountain bike, gloves will protect them from deterioration due to sweaty palms.
I ride two bikes with rubber grips, one always with gloves and one without. After a few years, the grip shifters on my hybrid are ruined and the rubber can't be replaced without replacing the whole shifter. The other bike's grips still look great after 8 years.
Trying not to mention what has already been said but:
Gloves that wick well will also help to keep you cooler (and cut offs are not a significant disadvantage in this sense, due to surface area to volume ratios and blood flow.)
The main reasons for choosing cut off gloves as opposed to more protective full finger gloves (given that both can keep you more or less equally cool if made out of the right materials) is partially personal preference, but often relates to better feel when dealing with zips, packed food/energy, gear adjustment and not having to take them off when working on the bike, some also claim a superior feel for the brakes.
What other activities are they useful for: potentially anything where you grip anything but don't need your finger tips covered or are willing to take the risk of finger tip injury, also for typing in the cold. I use gym gloves for cycling, better value, fit AND durability (in my case) than cycling equivalents at twice the price. I only use actual cycling gloves when it comes to full-finger + extra protection, or waterproof + breathable (since those are expensive in any sport, so I may as well have something with more attention to cycling relevant details).
