Either you will use a background thread to check your server for changes every once in a while(basically polling from server), or you use push notifications (whenever something changes on your server side, server will push the data that changed to you). Preferably push should be the way you choose cause it is both user and battery friendly. Also eases your server load.
In android, push notifications implemented mostly by using GCM. You can read rest through here http://developer.android.com/google/gcm/index.html
For a local network, you can use MQTT. There is an excellent blog post about it here http://dalelane.co.uk/blog/?p=1599 .