ActiveMQ is a free and open-source message broker that Apache maintains.
ApacheMQ is based on Java. It uses the Java Messaging Service (JMS).
It supports both producer/consumer and publisher/subscriber patterns.
It uses queues to keep and deliver messages in the order they were produced.

IBM MQ is a licensed message queue from IBM.
You can use the free but limited level of it for development.
You have to purchase the Advanced level to enjoy it fully.
IBM MQ queues work with queue managers.
They support both producer/consumer and publisher/subscriber patterns. 

Memphis is a free and open-source message broker that is maintained by the Memphis team.
Memphis comes with consumers, producers, and stations.
Stations are like queues.
Memphis implements publisher/subscriber pattern by leveraging consumer groups.
That is, by grouping consumers to the same group, Memphis emulates subscribers subscribing to a given topic.