"The apex, in racing terms, is the point on the track where the car is closest to the inside of a corner."
The racing line is a theoretical line which defines the position the driver should strive to achieve to maximise their potential to take a corner as quickly as possible. Be aware that this racing line will vary depending on many track conditions and driving situations. These include weather conditions, car class and driving style. So why is it important to understand where the apex is, other than knowing it is where you should be?
When a driver knows they have reached the apex, they can be confident that they can reduce the amount of steering angle applied and start increasing the throttle. i.e. they are 'coming out of the corner'. As the racing line only states that one point on the corner will be the time that the car is closest to the inside of the corer. It doesn't highlight whether this is early or late in the corner. This is where early, geometrical and late apexes are important.
The point of the apex will define where the steepest part of the corner is. In an early apex, the shallowest part of the corner is on the corner entry. If the driver drives an early apex corner when they are trying to out brake another driver. The shallower entry will mean a straighter car, which will mean the driver will be able to brake later in a straighter line. The downside to an early apex is it'll be much harder for the driver to get on the throttle quickly as the exit corner angle will be steeper. Thus the car will be using most of it's grip for latitudinal forces. In contrast to this late apex cornering is preferred for quicker lap times as the driver can apply the throttle sooner as the exit curve is shallower, therefore the steering wheel will be straighter.
Longer corners usually have two distinct apexes. The reason for this is, by definition, you are closest to the inside of the track, twice in the same corenr. On entry and on the exit. This is also know as a double-apex corner in racing.
Different corner types
Figure 1. 'Missing the apex' shows us how the orange driver is too far away from the apex of the corner. This is forcing the driver to drive steeper angled corner and therefore having to reduce his speed accordingly. In Figure 2. 'The Late Apex' the reference lap driver takes a wide entry into the corner and stays tight to the apex kerb on the latter half of the corner giving them the best opportunity to get on the throttle and exploit the straight run after the corner.