The act of casting can be done by all character classes, and a wide variety of monsters and NPCs. It simply means the time between initiating the use of a skill or ability until it is released. A cooldown period generally start after a casting, preventing the caster from using the same skill again for a short while.
Casting is often near instant in the Diablo games, but a some skills have a slightly longer casting time. While casting, the character is generally immobilised, and if hit before the skill is completed, it's usually cancelled without the cost of mana (or other resource).
Stronger spells generally have a longer casting time. The time between starting to prepare a skill and when it is executed is called "casting time". During this time it can be interrupted by other skills or abilities.