There are many types of natural clays on planet earth, and they are all beneficial in their own way: for either cosmetic and consumable use, or industrial use (think of bricks, tiles, porcelain and pottery).
When we talk about cosmetic use, we mainly use the clays that can adsorb or absorb, and sometimes do both.
Adsorbing means its molecules can dominate other molecules, making sure they don't enter the body and form a layer of protection. Think of water and oil, they don't mix together, but oil lies on top of water, meaning their molecules form a layer so they can't mix. Absorbing means that molecules can expand and engulf others (or be engulfed, like clays are engulfed by water), which changes their characteristics. This is how some clays can help detoxifying your body from harmful chemicals and impurities.
Beneficial clays you should try:
The most well-known cosmetic clay is called Bentonite Clay, of which you can read more here.
Moroccan Clay and Kaolin Clay can help balance oily skin and unplug pores, as they absorb the access oils on your skin.
Kaolin Clay isn't as absorbing as others, making it a great clay for sensitive skin.
Fuller's Earth Clay is wonderful for very oily skin, as is heavily absorbs oils. It can also be use as a stain removal.
French Green Clay has anti-inflammatory properties and are wonderful for people with sensitive skin.
Rhassoul Clay is amazing for aging skin, as it revitalizes, hydrates and helps your skin get firmer.