
Grip is a wonderful thing.
It keeps you on the road, it allows you to accelerate, turn corners, and then stop again. However, there is a limit to tyre adhesion and this, sir, is the limit of your car. It's vital to have a grasp of the basic science behind grip to help you drive on the limit.
Let's think about the contact patch of a tyre - this is the rubber which is actually in contact with the ground at any one time when you're driving along.
Diagram 1: The contact patch of a tyre is relatively small

This little area of rubber only has a finite amount of grip which it can supply to you. This finite level of grip can be split up into the front / back (longitudinal) and side to side (lateral) directions (see Diagram 2). Longitudinal grip is used up when you're accelerating or braking, and lateral grip when you're steering.
Here's the key point - if you are using up all of your available grip in one direction (e.g. braking), you will not have any available in the other (e.g. steering)
Diagram 2: Contact patch showing the direction of a. longitudinal and b. lateral force

Good racing drivers know how to get the balance of all these forces right, especially when cornering, but also when accelerating and braking. To maximise your available grip ensure you steer, accelerate and brake as smoothly as you can, and it's essential to avoid hard acceleration and braking when in a corner.
As usual, life is not quite as simple as it first appears, and the amount of grip available also depends on a number of other factors - one of which is the amount of weight acting on the wheel. If you decrease the amount of weight acting on the contact patch, this artificially lowers the amount of adhesion available, and vice versa. If you're accelerating hard, you will have less grip at the front end, more if you're braking. Photo 1 (below) shows a dramatic sideways weight transfer which has caused the inside rear wheel to lift, leaving no grip available at all.
Please see the dedicated article on weight transfers for more information on how traction is affected by driver input.
Photo 1: A front engine, front wheel drive car cornering hard

The best way of maximising your car's traction is to drive using smooth driving techniques. The way you change gear and use the throttle, brakes and steering can dramatically increase of decrease the amount of traction available. The best racing drives are always the smoothest, and this mantra applies equally well to all driving disciplines.
Grip can be increased in a number of ways, all of which are relatively easy to do. Adding aerodynamic down force such as spoilers, and splitters to a car can increase the amount of available grip by diverting the force of high speed air to your advantage. But beware of cheap spoilers which can be designed as a styling feature only - and if you are planning on fitting an aftermarket spoiler to your car ensure there is enough strength available in the bodywork as some key areas may need reinforcing.
Secondly, swapping your standard tyres for more specialist rubber can increase grip in certain conditions. For example, softer compounds usually provide better adhesion on racing circuits, however they usually wear out quicker - it's all a matter of compromise, so pick tyres which best suit your overall usage.
Your tyre is only 50% of the traction equation when driving - the other 50% is made up of the surface of the ground. Dry tarmac provides dramatically more grip than when wet, so ensure you adapt your driving style to suit the conditions.