There are three main ways to understand how a car is behaving.
The first and most complicated way is to calculate it. There are numerous simulation programs that can be used to calculate how a car react. These programs are as exact as the data that you put into them. For the most part any time spent using these programs offer the clearest way to see how different changes will effect the vehicle and to what degree they will effect things. These changes or tweaks can then be numerically optimized to create a very close approximation for production.
Generally however this method is saved only for car companies with a large engineering force.
The second method is to actually drive the car. By driving the car you can tell many things about how a car is acting very quickly, this helps isolate problems and gives a good general feel as to how a car handles.
The third method is a hybrid of the previous two methods. By measuring various things around the car as it drives you can collect data that helps you gain an even further understanding into what exactly is happening in the real world. This process is called data acquisition.
I will split data acquisition into 3 main components. The first will be the sensors that take various measurements and translate that data into an electronic signal. The second is the recording device that translates all of those signals into numbers. And the third is the computer and programs that help the user interpret those numbers.
For sensors there are a wide variety that can and are used by everyone from race teams to vehicle development teams to sensors found on every modern car.
As far as recording data acquisition for performance goes however the ones of the most interest are as follows.
Accelerometers: These lovely devices measure acceleration, and can measure the overall vehicle acceleration in the xyz axis, as well as measure angular acceleration (the rate at which a vehicle changes direction) along those same axis. They can even go as far as measuring individual wheel accelerations as you drive over all of the little bumps in the road.
LVDT/RVDT: linear variable differential transformer, and rotary variable differential transformer. These devices quickly and easily measure linear displacement and angular displacement. They are good for measuring everything from suspension displacement, to throttle/brake/clutch position, to steering wheel position.
Thermocouples: These sensors are electronic thermometers. They keep an active eye over various systems including coolant temperature, intake and exhaust temperature, oil temperature, and even tire temperatures if you use them when a car stops.
Hall effect: The hall effect sensor sends a electric pulse every time a magnet or a piece of steel passes by, by recording the frequency of these pulses you can record how fast something is spinning. These sensors are usually used for timing the engine, and for individual wheel speeds.
Infra red: Infra red sensors are another way of measuring temperature, only without contact. They are extremely useful in measuring tire temperatures while the vehicle is in motion.
Cameras: A camera allows you to see things that would be otherwise impossible to observe, though not accurate they can be used to help troubleshoot problems and as a general overview of what a car is doing on the track at any given time.
Strain Gauge: A strain gauge is a tiny circuit which is glued onto a part, it measures how much that part stretches/compresses. Knowing a parts strain and overall geometry you can calculate the forces going through the components and thus you can optimize the size and or shape to minimize the weight of the car.
For recording devices, there are many options on the market that range from tiny built in chips that exist in the sensors themselves, to very expensive and elaborate systems that record dozens and sometimes even hundreds or thousands of sensors at once.
Generally, most recording devices are limited in how much data they can take per second. If you did end up getting a system that can record one sensor at 100 Hz (100 samples a second), if you add on 9 more sensors chances are each sensor will only read 10 Hz (10 samples a second). While for some sensors 10hz may be overkill (like temperature), others 10hz is not even close to enough (like lvdt's on suspension components).
In the end however, the more expensive the system the more it will be able to record.
Finally, once you have all the data, displaying the data in an understandable form may range from anything as simple as an excel spreadsheet and table, to a complex program that links video feeds to individual sensors so you can see exactly where the car is, and what things are doing/recording.
Well, that quickly sums up data acquisition. If you have any questions please ask :)