Here is a short video showing an example app I created. I'll describe how to do the wiring, write an arduino sketch that can accept basic commands and send them and how to write the app itself. I asume you're already familiar with some of the basics, you made a few arduino projects and are familiar with the arduino IDE. If you have attempted serial communication with the arduino board everything should be fully comprehensible. If you have any problems or questions feel free to ask them in the comments section.
...and here is an example of receiving the data from the arduino board on the mobile. The temperature sensor connected to the arduino board is a ds18b20 sensor. Just to show how awesome app inventor is I added text to speech functionality - the app says the temperature every 15 seconds. The arduino code and the app is described in more detail in steps 4,5 and 6.