The majority of the workings of the human ear are muscle, and, just like any other muscle in the human body, it can be trained to work even more efficiently. Your ear is a marvellous organ - the human ear can identify literally thousands of different sounds.
For some people, playing the piano by ear comes quite naturally - they know as soon as they hear a particular series of notes or chords how that combination was played and where the persons hands were on the keyboard. By hearing the same piece of music over and over again you will find it much easier to recognise. Think of it like training a dog - the more you shout a command at the dog, the more it gets used to hearing it and what the necessary response is.
If you got a (very patient) friend to sit at your piano for two days straight and play nothing but major thirds (for example C and E or F and A, etc), your ear would be trained for the sound. Then, the next time you are walking down the street and you hear a car horn, I bet you would immediately identify that sound as a major third (this is because the vast majority of car horns are "factory-tuned" to a major third).
It can be done, and it can be done relatively easily. Once your ear is actually trained to identify these certain sounds, you'll be amazed at how much you recognise them as they occur everyday life all around you. Train whistles, car horns, even tannoy announcements! If you can identify a train whistle as producing a tritone portamento descending, then do you see how easy it would be to learn to play the piano by ear?
It can be difficult to find a friend who doesn't mind playing the same chords over and over again whilst you listen, so the best solution to this that we have found is get hold of a tape recorder, and record the music. If you can find a recorder with a numerical counter on it, all the better, as you are then able to rewind to any specific spot you like. This is really the initial step - you have to listen to the sounds in order to be able to recognize them.
When teaching, most piano teachers prefer to begin with melodic intervals such as skips of a major third or a perfect sixth, so this is a reasonable place for you to start also. Chord progressions can be more fun than the melodic intervals (and will keep you a little more engaged).
Training your ear to recognize specific sounds and thus transferring them onto the piano is a time-honored tradition that has served many people well in the past. It can be done, and it can be done relatively simply. There's no reason that you cannot do it yourself - learning to play the piano could not be simpler!