Playing by ear simply means that you hear a melody and use trial and error to find the notes used to create it. If you're already somewhat familiar with music theory, then this could be a good method to get started with. Try to pluck out simple melodies in the beginning, then try to create chords. This method allows you to explore the piano in a free way, and become acquainted with the instrument at your own pace.
Some teachers will teach you piano for a price. Private lessons are usually taught in your home or the teacher's home. The advantage to private lessons is that you get feedback from a real person regarding your progress. A teacher can also tell you where your weaknesses are, and how to correct them.
Thanks to advancements in technology, there are now piano lessons offered through the Internet. You can find many lessons at no charge, and they range in skill level from beginner to expert. However, unlike a piano teacher, the online lessons only go one way, and you can't get feedback on your progress or problem areas.
At many music shops, as well as online stores, there are DVDs available for purchase that can instruct you on how to play piano. These videos range in skill from beginner to expert level. The obvious advantage is that you can learn at your own pace, and if you get lost, you can pause the video and start over. There's also the advantage of not having to have a high-speed Internet connection, like you would need with Internet videos. However, there is no way to receive positive or negative feedback.