Is it possible to run .APK/Android apps on iPad/iPhone devices?

The app can't be run natively, but it could be run on an emulator. You can use ManyMo to embed them in a website and make users add your app to their home screen. This link should be useful for making the app more realistic. Users could then only press the share button and add the app to their home screen. All data will be deleted when the "app" is closed in their iOS devices so you should use the Internet/cloud for storage. It can't access camera or multi touch, but it may be useful.

Apple users can download your .apk file, however they cannot run it. It is a different file format than iPhone apps (.ipa)

There is another option not mentioned previously:

• Pieceable Viewer has unfortunately stopped its service at December 31, 2012 but open-sourced its software. You need to compile your iOS application for the emulator and Pieceable's software will embed it in a webpage which hosts the application. This webpage can be used to run the iOS application. See Pieceable's for more details.

It is not natively possible to run Android application under iOS (which powers iPhone, iPad, iPod, etc.)

This is because both runtime stacks use entirely different approaches. Android runs Dalvik (a "variant of Java") bytecode packaged in APK files while iOS runs Compiled (from Obj-C) code from IPA files. Excepting time/effort/money and litigations (!), there is nothing inherently preventing an Android implementation on Apple hardware, however.

It looks to package a small Dalvik VM with each application and targeted towards developers.

See iPhoDroid:

Looks to be a dual-boot solution for 2G/3G jailbroken devices. Very little information available, but there are some YouTube videos.

See iAndroid:

iAndroid is a new iOS application for jailbroken devices that simulates the Android operating system experience on the iPhone or iPod touch. While it’s still very far from completion, the project is taking shape.

I am not sure the approach(es) it uses to enable this: it could be emulation or just a simulation (e.g. "looks like"). The requirement of being jailbroken makes it sound like emulation might be used ..

See BlueStacks, per the Holo Dev's comment:

It looks to be an "Android App Player" for OS X (and Windows). However, afaik, it does not [currently] target iOS devices ..

YMMV