Wizards may not wear armour, as the close proximity of large masses of metal inhibits their ability to use magic. Therefore, as a general rule, wizards don't wear armour and for this reason the option is not included in the army lists. However, barding for horses is allowed where appropriate, see the army lists for details.
There are some exceptions, the most common being Chaos Armour worn by Chaos Sorcerers. Similar, though less common, is Skaven Warpstone Armour. Other exceptions are mostly associated with special characters, such as the Armour of Nagash.
A character who gains magical abilities thanks to a magic item can have armour if it is normally permitted by his army list. In this case the character's magical ability is vested in the magic item, and the presence of metallic armour does not affect his powers.