You need:
- an MQTT broker which runs on the server side. This could be something like the mosquitto broker which is Open Source.
- an MQTT client library which you include in your Android app, enabling your app to connect, subscribe, and publish messages. There are a number of Java options listed at http://mqtt.org/software
- to come up with a way of uniquely identifying users or devices, and then use that as a topic so that you can individually publish a message to one device.
Dale Lane has written a nice guide on this (http://www.dalelane.co.uk/blog/?p=1599) and there is also a good set of resources about MQTT (https://github.com/mqtt/mqtt.github.io/wiki)