I’ve seen pictures of MLB home plate umpires back in The Day holding what looks like a big black pillow. Presumably this was to protect them from wayward pitches.
Nowadays all they have is their face mask and no “pillow.”
When and why did they stop using that? And what was it called?
TIA
Chest protector. They are now worn under the shirt and have been for years.
IIRC, the external chest protector fell out of favor in the late 70s/early 80s, and was replaced with today’s internal protector. I remember reading in, I believe, “The Umpire Strikes Back” about the change-over. They stopped using it because the internal protector is more comfortable, and allows for more umpires mobility. You try to crouch down behind the catcher wearing one of the old ones, and you’ll see why they switched! 
The outside chest protector, usually referred to as the “balloon protector” was just used by American League umpires. National League umpires, starting in the 1930s and 1940s, just used an inside chest protector. In the late 1970s, both leagues adopted the inside chest protector as the standard although older umpires were still allowed to use the balloon protector.
I forgot who the last AL umpire to use the balloon protector was. Of course now, there are just umpires and they work games in both leagues.