The Z-score is an indicator of how many standard deviations a data point is above or below the mean. If the value is negative, that data point is below the mean, and if the value is positive, that data point is above the mean. For instance, a Z-score of 1 means that, that data point is 1 standard deviation above the mean, and a Z-score of -1 means that data point is 1 standard deviation below the mean. The reason it is called a Z-score is because the distribution from which it is based (the normal bell curve) is also known as a Z-distribution.
The most prevalent application of the z-score is in the z-test, which is often used to create the distribution for standardized testing. You can use the z-score to let individual test takers know where in the distribution of all test takers they lay in respect of the mean.