If a body of data is normally distributed (i.e., follows the bell-shaped curve), we can convert an individual's z-score on a given variable into a percentile. A percentile refers to the percentage of sample members an individual stands above on the variable.
If we go to this website and look at the bottom diagram, we can see how z-scores translate into percentiles (again, as long as the distribution is normal). For instance, we can see that a z-score of +1 (shown along the horizontal axis as 1 sigma) places someone at roughly the 84th percentile. Fifty percent of the sample lies below mu, and another 34.1% lie between mu and mu +1 sigma, thus adding up to 84.1.
On this webpage, you can see how z-scores and percentiles correspond to areas under the normal curve. There is also a website where you can simply type in a z-score and get the corresponding percentile, and one on which you can drag the z-score horizontally and see the percentage under the normal curve.
This photo of the board from a recent class meeting (thanks to Kristina) summarizes some of the major properties of z-scores.
