Arcane spell failure is the chance that a spell fails and is cast to no effect because the caster's ability to use a somatic component was hampered by armor. Divine casters do not suffer this form of spell failure.
Bards and warlocks have the Armored Caster feat, which means they can ignore the arcane spell failure chance for light armor when casting spells or invocations. Taking the Battle Caster feat allows them to wear medium armor without spell failure.
In depth Edit
Most armor provides more or less arcane spell failure. Some armor reduces the amount of failure taken, while others may even increase it. For example, most Full Plate armor has a 45% arcane spell failure chance. Shields vary from a minimum of 5%. Armor made from mithral or zalantar will have a reduced spell failure amount.
For the arcane spellcasting classes Wizard and Sorcerer, these are very important. Most of the time they cannot get away with wearing armor of any kind, even if they had the correct feats to do so. Bards and warlocks usually only ever wear light armor, and no shield, for the same reasons.
However, multiclassing with an arcane spellcasting class, with light or medium armor, can mean the extra AC is better than the small chance of failure.
The Still spell metamagic feat removes the arcane spell failure check. At +1 level, multiclassed arcane spell casters, especially sorcerers, may be able to get away with even full plate and tower shields if they can stand the lack of their highest level spells.