If you want to learn to play an instrument and you have no interest in learning the music theory required to sight read, then it's more than possible to learn how to play by ear. What you do need before you start however is some sensitivity for pitch and rhythm so that you can identify the intervals between notes and the speed at which you should play.
Can you sing? Can you clap along with a song on the radio? If so, then you should eventually be able to play by ear at least to some degree. When learning to the play the piano you traditionally start with learning how to read music first, but think back to when you learned how to read, couldn't you already use your instrument? Your voice? Of course you could. Piano teachers want you to sight read because it will allow them to teach you how to play from manuscript music, but if you can play by ear, you possibly have no need for the manuscript. This means that learning how to play by ear will allow you to teach yourself rather than have a tutor if you have the dedication to not only learning the skill, but also mastering it.
Listen to a piece of music, any piece of music. Hum along the melody line. Capture the right tone for the first note of the music. This is the note from which all the rest of the tune will come so you need to spend time getting this perfect. Of course you can play it in a completely different key to the original by starting on a different note and just following the line of the music up and down the keyboard keeping the intervals between notes the same as the music that you're trying to copy, but it won't sound the same unless it's in the same key, so take the time to locate the right note to start on.
Next you should work through the music one note at a time, tone to tone, working out the right intervals between each of the notes. If you have some idea of basic scale progression and can hum up and down scales you'll find this easier so this is something when you can practice when you're not at your piano. If you are at the keyboard, press any key and practice singing up and down the octave. Check if you have each note right by pressing the next tone before moving up/down the scale.
Once you've mastered the melody line, listen to the music again and play alongside it - this way you'll learn the rhythm of the composition. You'll know when there are intended rests and when notes are held. This is all part of the information supplied on a piece of manuscript but by practice you can train your ear to hear those slight pauses or learn when to play extended notes.
With the melody and tune in place, now listen to the original again and in the same way that you learned the melody, it's time to pick up on the harmony. Try to shut out the familiar melody line, and work on identifying the harmony. This may be more difficult to isolate but once you can recognize it you'll find it easier to work on. The more you know about chords, the easier it becomes, so learn all you can about the main chords in any key (also called "primary chords").
Learning to play by ear is a process of fine-tuning your ear to hear not only the dominant melody, but also the subtlety of harmony. As with sight reading, playing the piano by ear is something that comes over time with practice.
Duane Shinn is the author of a free newsletter on piano chords & chord progressions available at "Learn Piano"