In general, you might like to have a look at the format, also the assignment in while ($_=<DATA>)... is superfluous - while (<DATA>)... does exactly the same thing.

As to the exact problem, using strictures would have given you a clue to solving the problem since, assuming you rewrite $x =~ /^\[Note\](\w+)/; to my $x =~ /^\[Note\](\w+)/; (to remove the undeclared variable error that would otherwise ensue), you would have seen a use of uninitialised value in pattern match... error, the solution to which is (my $x = $_) =~ /^\[Note\](\w+)/;.

Having done all that, you can then go on to solve the problem of extracting the value from the field - maybe along the lines of $x =~ s/[^,]*,//;, so the final code should, IMO, look like:

<p>#!/usr/bin/perl

use warnings;
use strict;

while (<DATA>){
    (my $x = $_) =~ /^\[Note\](\w+)/;
    $x =~ s/[^,]*,//;
    print $x;
}

__DATA__
[Note]note_values,notes_values2
[Book]novels,magazines
[download]

Which generates:

$ perl tst.pl
notes_values2
magazines
$
[download]

As required. It then becomes a trivial exercise to ignore lines of no interest...