I wrote the book on Point and Shoot Nature Photography (no...I really did. You can find it on Amazon.)
A Point and Shoot Camera is one that was designed to be used in Auto or Program most of the time. It will, today, have intelligent Auto (scene selection auto), and lots of Scene Modes built in. Generally they have a fixed zoom lens. 3x to 83x (with the Nikon P1000 at 125x or something like that). They are generally more compact than any equivalent DSLR or even Mirrorless outfit. Some are still pocket-sized.
Yes they are still made. The old 3 to 4x zoom, pocket-sized P&S are gone, replaced by phone cameras, but most brands still have a 10 to 15x pocket sized offering...there are even 30 to 40x pocket sized P&S travel cameras. Most have small sensors...bigger than a phone but still 5.7 times smaller than full frame. They use contrast detection auto focus (for better and worse).