What is the "delivery mode" in AMQP?

Messages marked as 'persistent' that are delivered to 'durable' queues will be logged to disk. Durable queues are recovered in the event of a crash, along with any persistent messages they stored prior to the crash.

delivery_mode in AMQP determines if message will be stored on disk after broker restarts. You can mark messages as persistent - by seting delivery_mode property = 2 when you publish message for instance in PHP (PECL AMQP extension):

$exchange->publish($text, $routingKey, null, array('delivery_mode' => 2));

You would also need to declare queue as durable (or it will be dropped after broker stops)

$queue->setFlags(AMQP_DURABLE);