This is one of those things that can be difficult to search for if you don't already know where to look.
[ is actually a command, not part of the bash shell syntax as you might expect. It happens to be a Bash built-in command, so it's documented in the Bash manual.
There's also an external command that does the same thing; on many systems, it's provided by the GNU Coreutils package.
[ is equivalent to the test command, except that [ requires ] as its last argument, and test does not.
Assuming the bash documentation is installed on your system, if you type info bash and search for 'test' or '[' (the apostrophes are part of the search), you'll find the documentation for the [ command, also known as the test command. If you use man bash instead of info bash, search for ^ *test (the word test at the beginning of a line, following some number of spaces).
Following the reference to "Bash Conditional Expressions" will lead you to the description of -ne, which is the numeric inequality operator ("ne" stands for "not equal). By contrast, != is the string inequality operator.
You can also find bash documentation on the web.
test and [)-ne is under "arg1 OP arg2")testThe official definition of the test command is the POSIX standard (to which the bash implementation should conform reasonably well, perhaps with some extensions).