Hunting Animals
You can hunt animals with a weapon or a snare. All weapons work on all animals including predators. Snares only work on rabbits.
Animals can get scared and run off for several reasons. You get too close, they get wounded, or your shot lands close without hitting them. The sound of shooting your rifle or distress pistol can also scare them off (bows don't make a loud enough sound).
I don't recommend hunting big game unless you're ready to chase it down once it's wounded. They're fast, and it can take a while for them to bleed out and die. It's best to go hunting when your load is light and there's plenty of daylight left.
Weapons
Most of the time, you will not bring down any big game with one shot from a weapon. It's possible, but unlikely. Increase your chances by shooting it in critial areas (like the head).
Snares
Snares can be used to trap rabbits. Place snares in known rabbit areas then check on them in 12 hours. Then keep checking on them until you catch one. There's also a slim chance the rabbit could escape and leave your trap useless until you reset it. Snares can be found or crafted.
Tracking
When wounded but not killed, an animal will eventually bleed out and die but it may take a few hours depending on how badly they're injured.
You can track the wounded animal by following the blood trail on the ground. As they bleed out they may slow down but it takes a while. Once you find the corpse, you can harvest it. If you lose the animal, you may still stumble on it sometime later.
Note: A wounded predator that ran off may still attack you again. Wolves are less likely to do this, but bears always will.
Animals
Rabbits
Rabbits are always killed with one shot from a weapon. But if you miss they never run far so it's easy to keep on their tail (pun).
Deer
Deer are very skittish and can be tough to track down after they run away.
Wolves
A wounded wolf may or may not attack you again. Keep this in mind when tracking them down. Fighting off a wolf with a hunting knife or hatchet when it attacks will also wound it just like a weapon, eventually killing it after it bleeds out.
Bears
Bears can never be killed instantly with one shot, even in the head. It takes several shots to bring one down. If they see you, they will charge if you hit them, unless you're using a distress pistol which causes them to flee.
If you got a good shot or two off and managed to escape, the best option is to wait for it to bleed out and die rather than attacking again and risking a mauling and wasting ammo.
Harvesting
Make sure you select the best tool when harvesting an animal. Typically, this is the hunting knife. Carcasses will also freeze if left out too long which makes it take longer to do the harvesting. If you build a fire near the carcass then it will begin to thaw, but you'll just have to determine if the added time to thaw is actually saving you any time overall.
If you activate (or "open") an animal carcass, it will disappear more quickly. Even if there's still stuff left to harvest. Make sure you have time to harvest what you need before clicking on it the first time.