How to display a block if a Boolean on/off checkbox is checked?
Generally, if you're wanting to enable or disable a block based on some parameters (and assuming this block is defined within a module of your writing), you'd use hook_block_list_alter(), test for your condition, and then either display or hide your block.

You can see how this works by look at the block module's implementation of it's own hook, block_block_list_alter().

For instance, if you wanted to hide something based on the current node:

MYMODULE_block_list_alter(&amp;$blocks) {
    $node = menu_get_object();
    $my_hiding_condition = isset($node-&gt;somefield[LANGUAGE_NONE][0]['value']) 
      &amp;&amp; $node-&gt;somefield[LANGUAGE_NONE][0]['value'] == TRUE;
    if (isset($blocks['my_block_id']) &amp;&amp; $my_hiding_condition == TRUE) {
        // Hide the block.
        unset($blocks['my_block_id']);
    }
}