Broken shards (those brown things that you can only hold up to 10 of, distinct from rune shards. Do not confuse) can only be used to fill an empty spot in one of your own Ancient Wonder's rune wheel, or be used as an ingredient for Magic Academy crafting (typically, crafts recipes belonging to AWKP instants)
Rune shards (those fancy colorful things) can either be used to maybe fill an empty spot in one of your own Ancient Wonder (which, upon failure, turns into a broken shard), or used in lieu of KP worth 10AWKP on your own Ancient Wonders, or 15AWKP if used on anyone else's Ancient Wonder. For the value of 15, it doesn't matter whose AW it is as long as it isn't your own.