A Z-score is an observation (a single data point) expressed in a standardized form as Standard Deviation units. If all observations are standardized in this way they will have a distribution which has a mean of 0 and a Standard Deviation of 1.
Standardizing data in this way is useful of applying the properties of the normal distribution. Any given Z-score can be looked up in a table to ascertain how likely it is to have occurred by chance (according to the normal distribution).