This is described in the "man" pages for Shells such as the Posix Shell and Korn Shell.
The single bracket version [ ] is a shorthand version of the unix "test" command and is described in outline terms in the Shell manual and in detail in "man test".
The double bracket version [[ ]] is a Conditional Expression and is described in that section of the Shell manual.
There is much overlap in basic syntax between the two. They have totally different syntax for boolean "and/or" conditions.

In your examples, both of them need mandatory double quotes round the string variable $z. Only the second example gave a syntax error, but both were wrong.

Code:

z=`cat T|cut -d, -f2,2`
if [[ -z "$z" ]]
then
       echo success
else
       echo fail
fi


if [ -z "$z" ]
then
       echo success
else
       echo fail
fi

Beware that the "cut" will misbehave if there is one field in the data file but no commas. Try it.

Always put string variables in double quotes and you will have much less trouble with syntax.