what are advancment points and how do you use them?

Advancement points are a type of currency you get when you battle a pet close to the level of the pets you're fighting. They're used to buy things for your avatar. When something costs AP to get, you'll see the little "AP" symbol in the upper right corner of the box the image is in. If you want to get something you click the money bag in the upper left corner.

I hope this helped!

at the current point in time, there's very little use for them (buying avatar styles (things like mouths, eyes etc.), buying generator design save slots, and buying party preset save slots is about it), (though in the future it is intended that there will be a shop specifically for them to be used. It is unknown what exactly would be in the shop at this time)

to use them for buying avatar styles, Vanillalotus covered it, for the others:
generator save slots: select 'save current design', there's an option button that says "add design slots", this will give you the option to buy more slots (warning, DON'T do this when you have an unsaved design open, as it reloads the generator. Open a new generator tab/window and just select a species so it'll let you click the button, or delete one you currently have saved and save the one that's open before doing it)

party presave, it's just the "unlock more presets" button to get the popup that lets you pick what you'll use to buy more