It's a truism in RPGs (and fantasy in general) that armour and magic don't mix (fair warning: that's a TVTropes link) - that is, wizards are capable of astonishing feats of magic as long as they wear nothing more protective than robes and a pointy hat.
This is, of course, done for reasons of balance, and often handwaved away by saying that the armour is restrictive and prevents the wizards from performing the complex gestures (also a TVTropes link) necessary to cast spells.
Speaking as someone who owns a haubergeon, this blanket ban on armour (for this reason) seems at best suspect, especially given the existence of cleric-type characters. I for one would rather wear mail than robes with massive sleeves if my life depended on waving my hands about in complicated ways.
- wizards draw upon magic that permeates the universe
- They learn spells through study and on special occasions, such as reading a grimoire
- wizards prepare spells by memorising them, and cannot cast spells without first preparing them - the memory of the spell is wiped from their mind when cast
- They can do all of the normal things for this genre - throw around fireballs, create walls of earth, fly through the sky, etc.
- Spells have power proportional to their caster's skill and knowledge about magic and how to use it efficiently. This can be represented by a stat like Intelligence.
- The God(s) have nothing to do with this kind of magic (think of it as arcane rather than divine)
- Casting spells requires the wizard to say the magic words and perform the ritual gestures (see D&D's Verbal and Somatic components, let's ignore Material components for now)
- Wearing armour massively reduces a wizard's ability to use magic
Given the above, why would a wizard not be able to use magic while wearing armour?
I'm looking for a sensible in-universe explanation as to why armour in general would prevent a wizard from using magic while worn.
To be clear, some armour types are restrictive enough to (theoretically) cause problems - we see this IRL with things like tournament armours, but we have pictures of archers shooting in plate harness, and properly sized mail imposes virtually no restriction on freedom of movement, so a blanket ban doesn't add up.
This edit might invalidate the text of some of the extant replies, but hopefully it shouldn't invalidate the concept of any of them and narrows the scope of the question enough to be reopened.
If you're reading this and thinking of an answer based on metal interfering with the flow of mana, or armour being cumbersome and preventing the wizard from making the necessary gestures to cast their spell, please stop and consider whether you're adding anything that the fourteen existing answers on these themes haven't already added.