Austrians absolutely love their Patschen (slippers, house shoes). I have literally never entered a residence without being offered Patschen (Hausschuhe in Standard German). Upon arrival, you are expected to remove your own shoes, and once in the house you can slip on felt, wool, or cotton Patschen, typically with a semi-rigid base.
The word comes from a verb meaning to lightly clap or slap, reflecting how your feet pitter-patter on the ground.
Many Europeans think that US Americans walk around their homes with their shoes on. And you know what? I think a lot of us do. It is very uncouth in Austria to not remove one’s shoes, so that’s where Patschen come in to keep you cozy on these cold winter days.
You know what we North Americans think is uncouth here in Central Europe? That everyone and their uncle wore the same Patschen, and they’ve never once been washed!
Heated floors are gaining in popularity in Austria, and this might make Patschen redundant. This is a trend in the Western world, but specifically in Austria there have been energy and home renovation subsidies these past couple years that encourage projects like heated floors. They reduce home heating costs and energy usage because they are more efficient than wall heaters. If your feet get too warm, you may have to slip off those slippers!
To remember the word, think pitter, patter, Patschen as you pitter patter along the floor in your Austrian house shoes. This word is not used outside of Austria.
We have a similar word in Hungarian: pacsker (patch-ker)
It's German, used originally by certain Swabian groups living here, and it is a prettier pair of slippers than these Austrian ones: knitted, often colorful or even ornated with flowers: https://cdn.teol.hu/2021/12/opu7zzki-yfYUWucKQ_BjkV1yN50EX6Wt_d8MEI8WKE/fill/1347/758/no/1/aHR0cHM6Ly9jbXNjZG4uYXBwLmNvbnRlbnQucHJpdmF0ZS9jb250ZW50L2NjZWRjOTg4N2FhZDQyNmRhZTZiMGE5YzQ0NmYyYjYy.jpg
In the Hungarian language it is mostly used in an idiome: "Feldobja a pacskert" (fehl-dobe-yah ah patc-kert) = "he throws the slippers up", meaning that someone dies.
My feet tend to be warm, so whenever I go into an Austrian home (or enter my own home for that matter), I politely decline to wear the Patschen that are so kindly offered and just go barefoot. Austrians are usually surprised by this--I hope I am not being rude--but I don't need them and would rather not wear house shoes that have been worn by dozens of others.