Some words have the same etymology, root, but mean different things, such as mysterious and mystical. What are some other pairs (or more) that fall into this category, and what exactly is this category called?
There are actually lots of these. They are called doublets.
My favorite example, which is not mentioned on that Wikipedia page, is the following:
Genre, gender, generic, and genus all come from the Latin genus meaning "kind/class" (although the word actually goes all the way back to Proto-Indo-European *gen-/*gon-/*gn- meaning "produce, beget, be born").
Just to point out, doublets are cognates within a single language.
My favorite pair is galaxy and lactose, which come from the Greek word for milk, gala. The genitive is galaktos. Apparently, Milky Way wasn't a weird choice of name at all...
The word to describe this situatino (about words) is cognate. 'Skirt' and 'shirt' are cognate because they have a word in common etymologically.
My favourite pair is "canon" (=rule) and "cannon" (=weapon). Both from the same Greek word (kanna) meaning a reed, probably from a semitic root. I think that is an awesome stretch of meaning.
Cognates with different meanings from different languages/dialects are known as False Friends.