Learning to corner on a bicycle effectively at low speeds will help you a lot when riding in dense urban environments. This comes up a lot in stop-and-go traffic as cars or other bikes cut in front of you, as you notice pot holes, etc. This skill is especially helpful if you have clipless pedals (or cages/toe straps) since it's harder to put your foot down quickly.
One way to think about turning on a bicycle is that you can decrease your turning radius (i.e. turn more tightly) by leaning the bike more. How much you can lean the bike before falling depends on how fast you're going and how much traction you have. When you're going really fast you can have a more aggressive lean angle; for instance, in MotoGP races you'll see motorcycles at high speed lean so far that the rider's knee will touch the pavement. When you're going slowly normally you cannot have an aggressive lean angle because the bike will tip over. This means that normally when you're going really slowly you can't pivot the bike around obstacles quickly.
If you lean too aggressively, the bike will start to tip over. If this happens you can usually correct the action by accelerating. As you accelerate it changes your angular momentum and the bike will right itself.
There's a really effective way to combine these two properties on a bicycle. Here's how it works. You stand up and get out of the saddle and lean over at an aggressive angle. By standing you increase the net torque on the system because you move your center of gravity out further from where the tires contact the road; this makes the turn more aggressive. Then as you are near the middle of the turn you accelerate by using as much force as you can to turn the pedals. Because you're out of the saddle, you should be able to get extra force when cranking the pedals because you'll naturally use your weight to help rotate the pedals. By accelerating like this the bike will right itself (and you'll gain speed).
If you do this properly you'll feel like you're "throwing" your weight around; first when you stand up and lean over, and then again as you use your weight to force the cranks around faster. The whole thing will quickly feel really natural as you get used to how far you can lean and how fast you can accelerate. This technique is easiest if you're going slowly enough that you can accelerate through the turn with a single rotation of the cranks because that will let you use your weight to rotate the cranks most effectively.
On a fixed gear bicycle this process is a little bit more difficult because you can't control the pedal position to stand or control the crank angle as you begin accelerating. I would argue though that this skill is more important to master on a fixed gear bicycle since foot retention is more important on fixed gear bicycles, and putting your foot down on a fixed gear bicycle is more annoying since you can't easily rotate the cranks to the right starting position.
One more note: be careful when employing this technique when the road is wet. When the road is wet you have less traction than usual, and if you've developed intuition for how much torque you can apply on a dry surface you can lose traction in the rain.