Single [] are posix shell compliant condition tests.
Double [[]] are an extension to the standard [] and are supported by bash and other shells (e.g. zsh, ksh). They support extra operations (as well as the standard posix operations). For example: || instead of -o and regex matching with =~. A fuller list of differences can be found in the bash manual section on conditional constructs.
Use [] whenever you want your script to be portable across shells. Use [[]] if you want conditional expressions not supported by [] and don’t need to be portable.