Carrier or base oils are the oils you’re probably most familiar with: olive oil, coconut oil, sunflower oil, and any other oil you might cook with. Butters are also fats, but tend to be solid at room temperature. Fats typically come from nuts and seeds, but some – like tallow, and lard – are derived from animals.
Like carrier oils and butters, waxes are lipids. Lipids are insoluble in water (their chemical structure is a long, non-polar chain that doesn’t like the mix with water’s polar molecules) but can be forced to combine in a process called emulsion. Because waxes and fats are both lipids, the line between them can be blurred. While jojoba looks and acts like an oil, it’s actually a liquid wax. Waxes can also come from plant (like bayberry and candelilla wax) or animal (like beeswax and lanolin) sources.
There are also non-organic fat-like substances like mineral oil and petroleum jelly. Unlike waxes and fats, these are derived from non-living things.