There are two most common ones. The Gibson style that goes right at the end of the fretboard and the Fender style that is kind of embedded in the fretboard (like yours).

Nuts of the same type are usually a similar enough size to work with. Sandpaper is the main tool for fitting a new nut. a slightly oversized nut can be sanded to size with a bit of patience.

Nuts are so cheap I'd buy two or three in case you mess up. Watch a few YouTube videos for guidance (there are many) like this one by Dan at Stewmac which is a pretty good introduction...

https://www.youtube.com/watch?v=Pi7LtmdOrFw

Fitting a new nut is a great DIY skill for any guitar player to learn.