The ability to choose the right line through a corner, how to get through the apex effectively, and a basic understanding of counter-steering are all key components to successfully and effectively wrapping around a corner.
When you approach the corner, never over-ride your line of sight. In other words, don't drive faster than you can see to stop in case of an emergency. Determine a safe cornering speed before you initiate the turn, braking hardest before you start counter-steering, then letting off the brake as you lean the bike in to the corner. By the time you reach the apex, you should be ready to start accelerating again.
Perhaps as much as 1/3 of all motorcycle accidents are single bike accidents where the bike ran off the road because the rider "just couldn't make the turn." In many of those cases, proper cornering technique may have played an important part in preventing the accident. Knowing how far your bike can lean in a turn is key to knowing how quickly you can make a corner. On todays sport bikes, if you're not scraping parts on the ground in a lean, you're not leaned over all the way.
Your tires are the only thing that separates you from the pavement. On an average sport bike, there are only a few square inches of rubber "contact patch" for each tire. Good grip is determined by things like tire condition, road conditions, dynamics, and sudden inputs.
You don't have much holding you up, so every aspect of grip is important and must be considered when riding. If you take even one aspect of grip out of the equation, you put yourself at great risk of falling. You can't control the road conditions, but you can keep yourself in the area of the road where more grip is available. Riding in the tracks from car tires will usually provide you with the best traction. This is because the car's tires are constantly cleaning the surface as they roll over it. Try to keep your bike in the area of the road where the car tires typically travel.
Some of the worst times for traction are the first few hours of (or after) rain, when oil and debris floats to the top. Extra care should be taken to eliminate sudden inputs, sharp turns or hard braking/acceleration when conditions are poor.
Unlike a car, you only have two tires on your bike. If you lose one tire due to a flat or because you run into dirt, oil etc, especially in the front, you will almost surely fall if you're not ready for it. Don't take your tires for granted. Make sure they're in good condition, and keep them at the proper air pressure. When they're worn or damaged, replace them immediately.
The dynamics of a motorcycle related to the physics involved in keeping it upright are pretty interesting. They're also handy to know in case of an emergency. By its very nature, a motorcycle in motion is more likely to stay upright than when it's not moving, or when it's moving slowly.
The faster you're going, the harder it is to fall over, and the more violent it will be if you do. The reason for this is gyration. The major moving parts on a motorcycle, (tires and engine), create stability while in motion. If you are riding slowly and are 'bumped' by an outside force, less pressure is required to knock you over than if you were moving faster.
