You can use [`get_the_terms()`](http://codex.wordpress.org/Function_Reference/get_the_terms). I've adapted the following from an example on that page:

    $terms = get_the_terms( $post->ID, 'visits' );
    						
    if ( $terms && ! is_wp_error( $terms ) ) : 
    
    	$visits_name = array();
    
    	foreach ( $terms as $term ) {
    		$visits_name[] = $term->name;
    	}
    endif;
    						
    $terms_list = join( ", ", $visits_name );
    echo $terms_list;

**EDIT:**

Using [`wp_list_pluck`](http://codex.wordpress.org/Function_Reference/wp_list_pluck), as suggested by [Telos](http://wordpress.stackexchange.com/users/31073/t31os), is much easier:


    $terms = get_the_terms( $post->ID, 'visits' );
    						
    if ( $terms && ! is_wp_error( $terms ) ) : 
      echo join( ',', wp_list_pluck( $terms, 'name' ) );
    endif;