Generally speaking, Manifestations occur to those individuals who are Potentials - that is, who have a genetic predisposition to eventually gaining powers. Currently, the exact mechanism by which this occurs is not known - it is not possible to identify Potentials based on studying their genome or other factors. Thus the term "Potential" can only be applied retroactively, and has little practical use.
The Manifestation of powers is often sparked by a specific trigger. As most triggers are emotional or otherwise personal, the most common age at which they occur is adolescence. The process appear to be largeley mental - the exact power that appears is related to the personality of the individual, or the events of the trigger. For example, an individual (such as Evan Rogers) with a manipulative persobality may end up with the ability to control the minds of others; or someone whose power Manifests becaause they are jealous of the attention someone else is receiving (as in the case of Ashley Weber) may end up with an aura of glamor.