I got a free .mobi file today and it took a couple of minutes to get the file working in the Kindle app on the android. First of all, download the mobi file to the android. By default it will save it in the download file on the SD card. Then using your application of choice, move the file to the kindle directory. If you have the kindle application running, you will need to kill it and then relaunch it so that the kindle app restarts and realizes the file has been loaded.
I use the OI File Manager app for moving files on the android, and Advanced Task Killer to kill applications when required. The other android apps I use can be seen at my Absoblogginlutely appbrain page