The goal of neuropsychological testing is to measure certain functions of the brain, and make sense of an individual's behavior, learning, cognitive processing, and perhaps even potential to achieve. There is not one single test, but a "battery" of different tests (individualized to the needs of the person at hand) that tap into different aspects of brain functioning. For instance, there are measures that tap into lanugage functioning, measures that assess non-verbal functioning (visual/spatial processing), measures that assess immediate memory, longer term memory, planning, organization, the ability to integrate parts into wholes, etc.
In neuropsychology these measures are then compared to 'normative data', in other words, results from the population at large, in order to compare one person's abilities and difficulties to the "average person". In this way, we are able to produce standardized and reliable data that can help make sense of a person's abilities, deficits, and vulnerabilities compared to the 'general population'.
(Standardization means that the way I measure and record something is the same way that another clinician would, meaning that my data would be reproduceable by anyone else with the same training; and reliability refers to the idea that the tests measure what they're designed to measure.)
In my practice, I also emphasize a person's life history, their school based experience, oppurtunities, challenges, and other factors that could come into play. Essentially, my job is to reconcile the 'hard data' with the individual's history and presentation in the here and now to see the person as a "whole" rather than as a collection of numbers and diagnoses.
This information is then put into a report that can commmunicate the person's functioning to psychologists, psychiatrists, neurologists, therapists, as well as family members.