[[ ]] brackets are an extended version with a more "C-like" syntax, supporting all the usual operators as well as things like && for and instead of -a, || instead of -o.
[[ ]] is a Conditional Expression. Explained in the manual for your Shell in the "Conditional Expressions" section.
[ ] is a Test. Explained partially in the manual for your Shell in the "test" section then further in the manual for the "test" command.
Though there is much overlap in the syntax for basic conditions and tests, there are also major differences in the syntax for "and" and "or" etc. .