I'm new and was doing these Wizard Quests and I received this:

What in the world is this used for? When I clicked on it in my Inventory it said I could use it on my pet. What would that do? :/

It'll change your pet into a Dawn Torrey.

Oh, so is that what all "elixirs" do? Then do potions just turn your pet into a different colour but same species?

That's the gist of it. :) Potions change color, elixirs change color and species. The potions tend to be more expensive, especially since the most popular colors can't be found in elixirs.