This works with android phones and the HC-05 and HC-06 bluetooth modules, as well as pretty much any arduino or arduino clone.
Step 1: Connect Module to Arduino
Pretty straightforward, connect the bluetooth module to the arduino like so
Step 2: Plug in the Arduino and Upload the Sketch
Download the ino file and upload it to the arduino.
Step 3: Connect to the Bluetooth on Your Phone
Open up the bluetooth settings on your phone and look for the module. Mine is named Sailfish but yours should be named either hc-05 or hc-06. The password is almost always 1234, if not, try 0000.
Step 4: Connect to the Bluetooth Module
download a bluetooth terminal app from the appstore, this one works great: https://play.google.com/store/apps/details?id=Qwerty.BluetoothTerminal&hl=en and open it up. Open the app, hit connect to a device - insecure and select your bluetooth module from the list.
Step 5: Ready to Go!
Finally open up the Serial Monitor in the arduino IDE by hitting the magnifying glass in the top right corner, and you should be good to go. You now have a back and forth extremely simple connection between your phone and your arduino via bluetooth.