The caster level (or casting level) is a number separate from the player level determining a caster's skill in the arkane arts. It affects the power, duration and mana cost of some spells.
The caster level is not directly related to the main character level or to the level number given to individual spells.
The player's caster level is not displayed in the user interface, but it can be calculated as
(full_casting + full_mind) / 10
The player caster level is also always limited to between 1 and 10 (inclusive, barring cheats). The max cheat causes ALL spells to use level 15.
Generally, the caster level that the player had when the spell was initiated is the one that matters - the potency of active spells does not change if the player levels up or puts on magical equipment. Precast spells created by the player do not remember the caster level but instead use whatever the value is when they are invoked - so there is no need to re-prepare spells when your character levels up.