Attribute potential is a stat separate from each corresponding primary Attribute — it is not related to the character's current abilities, but serves to signify how easy it will be for him/her to improve. In practice, if a character's stat has reached its potential (which can be checked in the "@" or Alt-q screen), any effort taken to train it (as detailed in the appropriate article for each stat) may instead increase the potential by one — thus yielding no immediate benefit, but allowing more training to increase the actual stat. In 1.2.0, most if not all training methods (except direct increases from eating certain corpses) were removed, making stat potentials now a more differential aspect for races.
Potential is even more heavily influenced by race than natural attributes. Trolls will not only have very poor mental stats, but their potential for these will also cap at just double digits or even below (for instance); their Strength and Toughness potentials on the other hand are more often above 40 than not. Similarly, Gray Elves will often have Mana, Appearance, Dexterity and Learning potentials in the thirties or higher, but get Toughness potentials in the lower tens and Charisma possibly even lower. As a rule of thumb, expect your potential to reflect the attribute tendencies of the races in a multiplied manner.
Potions of potential <attribute> will only increase the potential as advertised, so they serve only to make training certain stats easier. Potions of gain attributes, as well as the enhancement potions for each individual attribute (as of v. 1.2.0), will only increase potential if it was already reached (since the potential can never be lower than the natural stat).
As of version r48, attribute potentials can be abused (which can cause them to decrease over time) if stats that are already at their minimum value would be lowered/abused.