A racing circuit is comprised of left and right turns and straights that link up to form an unbroken 'loop'. Pretty simple stuff really. Each racing circuit is of varying length, with some tracks up to 22m wide at points, like Malaysia, and others are only 6m in points, like Monaco.
The fastest way to get from point A to point B is always going to be a straight line. Thus, the basic key is to keep the car as straight as possible as often as possible. This typically, but not always, means finding the straightest line through a turn, or series of turns.
Also, remember that when the wheels aren't pointed straight ahead relative to the rest of the car, they're generating side forces instead of forward forces. This means they're basically wasting energy and decreasing the life of the tyre as they slide laterally over the rod.
Now for some terminology.
Braking point: This is the point where you apply the brakes and begin deccelarating for the corner.
Turn in: This is where you start turning the wheel proper to get through the corner.
Apex: This is the a very important component of a turn and is the point where the car reaches the inside of the turn and usually is right on the end of the track.
Track out: is the exit of a turn and usually found on the outside edge of the track.
You'll notice the above diagram defines early and late apexes. The "standard" apex is the yellow line and is in between the other two apexes. Typically you would use an early or late apex to attempt an overtake, or if it suits your particular style of being a late braker (late apex and sharper steering input, with a slower entry speed). An early apex might also be employed if you're overtaking or like to brake early, minimise steering inputs and maintain a higher entry speed with lower mid corner and exit speed. Where you overtake a car and the apex you employ to do so depends on where the car is. For example, you might overtake on the side by taking an earlier apex and "parking it" on the apex, making the other car effective slow more than they would. This is effective if they havent' also take an early apex, as it ruins their line and exit speed.
Now you know the basics of cornering, the tricky part is stringing it all together. At first, the best approach is to take the shortest line between two points that is the standard apex in the middle of the corner. You also need to allow for change of direction (transitions) such as driving through the Esses at Suzuka. In that case you may not want to go completely from side to side because you're going to be extendind the overall distance through the series of turns.
Once you're comfortable with the basic line, start thinking about your own prefferred driving style and how this effects your line and apex of a corner.
It's also worth noting that many tracks have green bollards on the corners. These aren't there just to hit. These are corner markers, used when the corner may be hard to spot. A good example of this is at a newer track like Abu Dhabi. Here the bollards are pretty much spot on the apex and you can aim your front inside wheel at them and just skim them as you drive through the corner. Certainly if you're consistently missing apexes and the corner has a bollard, aim for the bollard. It's what it is there for.
Finally I should mention braking. Theoretically all braking should be done in the braking zone and when the car is still going straight, before turn-in. Why? Again, think about what the tyre is doing. If the tyres are pointed straight ahead, they can be used exclusively for slowing the car. If you turn the wheel while braking, the tyre must both provide grip to slow the car and grip to turn the corner without coming off. You also run the risk of pinching or locking a brake and causing the car to spin. Trail braking can be used succesfully though. This is when braking is applied while turning, before the apex. Trail braking should be the gentlest of braking though. If you stab the brakes while turning, all you'll do is unsettle car and cause under/oversteer. Gentle application of the brakes while turning can help the car turn faster and hit your apex. It will also allow you to start braking later, as you can stop braking much closer to the apex. Generalyl trail braking requires left foot braking. See here for more detailed information on left foot braking.
One final note (I promise), now that I've mentioned trail braking: The Apex is the answer to the question "When should I stop braking and start accellarating. You should not accellarate before the apex or you run the risk of overstreet/understeer.