Can I install Google Play and run Android apps on my iPhone or iPad? I have an iPod Touch 4G (this is where you all start screaming at me). But I like the apps on Google Play. So if I downloaded a Google Play app on my computer and sent it to my iPod, would it work?
iOS and Android are two different systems. The only way to run Android Apps on iOS would be with an emulator.

There exists a project called iphodroid. It has a Google Code project page. The last update from the homepage is from May 2011 and given the rapid development regarding Android and iOS I doubt that this is still functional. You can try it, but at your own risk! I have never tested it.