You are closing your function right after the $args array. You need to enclose properly
function local_business_directory_register() {
$args = array(
'label' => __('Business Directory'),
'singular_label' => __('Business'),
'public' => true,
'taxonomies' => array('category'),
'show_ui' => true,
'capability_type' => 'post',
'hierarchical' => true,
'has_archive' => true,
'supports' => array('title', 'editor', ),
'rewrite' => array('slug' => 'businesses', 'with_front' => false),
);
register_post_type( 'businesses' , $args );
register_taxonomy("business-type", array("businesses"), array(
"hierarchical" => true,
"label" => "Business Type",
"singular_label" => "Business Type",
"rewrite" => true
)
);
}