How to Teach Yourself to Play the Piano

To learn the piano without a teacher (I.e by ear) will require a strong musical ear and patience...
First of all, you obviously need a piano, or a place with a piano that you are free to use in spare time for a few hours a week at least.
Make sure the piano is reasonably tuned (most pianos are kept in tune).
To learn to play the piano well on your own, the best possible first step is to learn your basic scales, most importantly the Major and Minor pentatonic. This doesn't require a teacher as the internet is full of information and resources for musical scales - just google it! ;)
It is important to learn these scales first - if you wish to play basic piano songs - and know what key the song is in. Nine times out of ten the notes will be in the scale of that key.
To learn your own favourite songs, just listen by ear and work out the right notes, once you have the melody down you can usually work out the chords using notes from the melody. If you can read music, resources can be found on the internet.
This is basically the furthest you can go on your own, but just by simply practicing your scales and training your ear, you can play anything if you put enough time into it.

Don't stress out, remember its just fun that way you will stay more interested.
It is a lot easier if you already play a melodic instrument, as you will have a familiar view on musical notes and will be able to work with your other instrument(s) for a guideline.
Practice scales as much as possible, as they are the basis of every song, and will greatly improve the speed of your finger movement.
As tempting as it can be, try to play without the sustain pedal pushed down as it will sound clearer and you will be able to hear your mistakes more. This can help you to improve a lot.

If you want to play piano as a career or seriously wish to go further into your playing, it is highly recommended that you do get a teacher, without one you will pick up many bad habits that will take years to unlearn, as with any instrument! It is very difficult to obtain a profession or high grade without an appropriate teacher/tutor.