A sentient being can only hold so much magic inside of it before it starts to spill. Part of this metaphysical space is the "soul" -- the part that lives on after death and can be captured in shards and some specially-designed positronic brains. Thus most living beings are limited in how much power they can carry around, and rely on external sources like artifacts, tools, and borrowing from deities to use more.
A wizard is a magic-user who has replaced some or all of their soul with magic -- and in doing so, lost the ability to be conventionally revived. Some wizards end up a being known as a lich as a result of this process, but not all lichs are wizards, and not all wizards are lichs. This often results in a relentless pursuit to be remembered. Death is cheap, and they've lost the way to come back fully from that.
This often manifests in quite a few ways. Some wizards carefully select spells, artifacts, and other abilities to murder on an unprecedented scale, to be remembered for sheer power. Others choose a gimmick and try to make their mark by being bizarre, helpful, or just plain weird.
Regardless of what philosophy any given wizard chooses, they all have the capacity to be extremely dangerous, and due caution should be taken when interacting with them, especially larger orders of wizardry.