It is not possible to run Android applications on iOS natively (which powers iPhone, iPad, iPod, etc.)
This is so because the two runtime stacks take completely distinct tacks. iOS executes Compiled (from Obj-C) code from IPA files, whereas Android executes Dalvik (a "version of Java") bytecode packed in APK files. There is nothing fundamentally stopping an Android implementation on Apple hardware, however, besides time, effort, money, and lawsuits (!).
Each programme appears to come with a compact Dalvik VM that is geared toward developers.
Look at iPhoDroid:
appears to be a dual-boot fix for jailbroken 2G/3G smartphones. There is not a lot of information, but there are some YouTube videos.
Look at iAndroid:
A new iOS programme called "iAndroid" for jailbroken devices simulates the Android operating system on the iPhone or iPad, or iPod Touch The project is taking shape, even if it is still very far from being finished.
I'm not sure what method(s) it employs to make this possible; it might be emulation or just a simulation (e.g. "looks like"). It sounds like emulation might be used given the requirement of being jailbroken.
As suggested by the Holo Dev, check out BlueStacks:
It appears to be an OS X "Android App Player" (and Windows). It doesn't [currently] target iOS devices, afaik.