I have written a REST web service in Netbean IDE using Jersey Framework and Java.
For every request the user needs to provide a username and a password, I know that
What back-end? If JAVA then you can use REST with Java (JAX-RS) using Jersey.
On the Android side you can use this simple RestClient to work with that REST service.
For JSON <--> Object mapping on both sides (Android, Java back-end) you can use GSON.