Do Android phone apps run on Android tablets? Apple iPhone apps also run on iPads. I don't know if the same is true for Android.

Do Android phone apps run on Android tablets? I figure the API is the same but are the two formats interchangeable at all?
Short answer: It depends.

Long answer:  

Android developers have the ability to specify requirements for their apps both in terms of software (minimum OS version, for example) and hardware.  For example, an app that does turn-by-turn navigation may require the Android device it's being installed on to have built-in GPS.  Such an app would be listed as "incompatible" with any tablet that doesn't have a GPS chip (the tablet reports its hardware features to the Play Store).  The same is true for a reverse situation, where an app that requires a minimum screen resolution may work fine on a tablet, but show as incompatible for phones that are below that limit.

Another example is Google TV. Since none of the current models have touch-screen support, any apps that require this feature would be listed as not compatible on the Play Store.

These limitations are only being enforced by the Google's Play Store.  This means that if you manage to side-load the app's APK file, it may work just fine on any device.  However, the lack of minimum requirements will mean that the app will be unstable.