Both cuts get shorter as they move down the sides of your head. The difference is where they finish. A skin fade takes the hair all the way down to bare scalp somewhere on the side. A taper fade shortens gradually but leaves a little length at the bottom, usually just around the ears and the neckline.
The skin fade
A skin fade gives the sharpest contrast between the top and the sides. It photographs well and it looks freshly cut for about a week.
The trade-off is upkeep. Because the bottom is bare skin, even a few days of growth is visible as a shadow. Most people who wear a skin fade come in every two weeks.
The taper fade
A taper is more conservative. The shortest point still has clipper length in it, so as it grows the shape softens rather than disappearing. It reads well in an office and it holds up for three to four weeks.
If you cannot get to the shop on a strict schedule, a taper is the more forgiving choice.
How to ask for it
Tell your barber two things: how short you want the lowest point, and how high up the head you want the fade to start. Low, mid and high describe where the fade line begins, not how short it goes.
If you are not sure, say so. Your barber can start conservative and take more off, but they cannot put it back.