Now, based on your full code, there’s 3 problems:
Line 60 of your code is missing a semi colon and throwing a fatal error. So:
$instance['description_check_box'] = $new_instance['description_check_box']
Should be:
$instance['description_check_box'] = $new_instance['description_check_box'];
On line 66, you’re using strings instead of proper boolean true or false values. Both if these are ‘truthy’ (see this answer to another question for what that means), so will both resolve as true if used in an if statement. if ('false') is true. So:
$description_check_box = $instance[ 'description_check_box' ] ? 'true' : 'false';
Should be:
$description_check_box = $instance[ 'description_check_box' ] ? true : false;
On line 121 you haven’t got on in quotes, so you’re comparing $description_check_box to the constant on. This should be in quotes if you want to check if the value of $description_check_box is on. So:
<?php if($description_check_box==on){ ?>
Should be:
<?php if($description_check_box=='on'){ ?>
BUT. This still won’t work. Because you’d be setting $description_check_box to true or false based on whether it’s checked on line 66, but later you’re checking if the value is 'on'.
You either need to set the value of $description_check_box to 'on', or change your comparison to checking if $description_check_box is true.
You should also use isset() before using $instance['description_check_box'], because otherwise you’ll get an undefined index notice in the customiser when WP_Debug is enabled.
Refer to my other answer for my approach, which avoids all these problems.