Wordpress - Custom Taxonomy as checkbox or dropdown

As of WP 3.7 (https://core.trac.wordpress.org/ticket/14206) you can add this argument to register_taxonomy:

'meta_box_cb' => 'post_categories_meta_box'

to get the built-in checkbox category style metabox without having to make your taxonomy hierarchical.

Also you could instead provide your own callback function to create your own metabox (i.e. with a dropdown).

You probably did not set the 'hierarchical' argument to true in your register_taxonomy. This would mean that it defaults to false, which gives you a tag-like interface.

Add 'hierarchical' => true to your register_taxonomy.