Mnemonics

"hex", "times"
-- all things hexadecimal
-- multiplication in grammar school math

Used alone

x used as infix operators perform replication

Operators containing this character

xx is also a replication operator. See above.

When used inside regexes

\x same as in an interpolated string (see below)

As an adverb

:x and :ex regexe adverbs perform repeated matches

Other uses

\x inside an interpolated string introduces a hex-encoded character
0x introduces a literal number in hexadecimal format
%x in a printf format outputs numbers in hexadecimal format