Restart your iPhone/iPad/iPod Touch. If you don't know how to do that: hold down the power button until "slide to power off" appears, slide, wait until the device shuts down, then push the power button again to power on.
If that doesn't clear up the problem, the next step is to delete the app and re-download it. You can follow the instructions here under "Reinstall the application."
If Moonlight Mahjong for Mac is crashing on startup, please launch the Font Book app (in the Applications folder) and make sure the fonts "Arial" and "Georgia" are present and enabled. These are system fonts, so they are supposed to be there, but for whatever odd reasons they end up being disabled on a few systems - if that's the case, you need to enable them for MM to work. This will be worked around in the next update.
First, make sure you didn't just misplace it. The home screen can have multiple pages of apps, so make sure you have checked the other pages. Better yet, just press the home button once or twice and do a search for the app you're looking for.
If you have accidentally deleted the app before syncing your device, you can download it again. When you try to download or buy it you'll get a dialog like "You have already purchased this item, do you want to download it again for free?" Just say yes at that point. See this article under "Reinstall the application" for details.
You can also control what apps are installed on your device using iTunes. Apps that you purchase on your mobile device are backed up on your computer when you synchronize.
Email midnightmartian @ gmail.com. Please keep in mind that the guy who does Midnight Martian loves to hear from you (usually) but is very busy making more apps and features.