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 look at page 26 of Naked Statistics, we can translate any particular z-score into a percentile (again, as long as the distribution is normal). For instance, we can see that a z-score of +1 (shown along the bottom of the diagram in the book as mu + 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.
This photo of the board from a recent class meeting (thanks to Kristina) summarizes some of the major properties of z-scores.