How to Play Jazz Piano

Jazz is an art form that has grown from its blues origins to draw influences from just about every genre of music there is. For the beginner, though, it's perhaps best to focus on early swing and learning to improvise. Here's a pretty easy way to get going!

Steps

1. Listen. This is by far the most important step in becoming a musician. Find as many recordings as you can get your hands on. Don't discriminate--listen to the old greats, like Art Tatum and Count Basie and Thelonius Monk, as well as up-and-coming pianists of today. Listen, take what they do, and apply it to your own playing. Doing this consistently will make you an excellent jazz pianist.
2. Assuming you already know some very basic theory, first learn all 12 major scales (there are twelve different sounding scales, but in theory B/Cb, F#/Gb and C#/Db are separate scales). Learning all the scales will be extremely helpful.
3. Make sure you can read music and can play some basic stuff, even if it's not jazz. The first real step in your journey will be to break away from "the dots" and train your ear. So...
4. Buy a songbook of one of the masters: Cole Porter, Gershwin, etc. Make sure that chord symbols or guitar tabs are written above the melody line, like "Dbm7."
5. Learn a major 7th (1 3 5 7), minor 7th (1 b3 5 b7), dominant 7th (1 3 5 b7), half diminished (1 b3 b5 b7), and diminished chord (1 b3 b5 bb7) of every key. So, for example, to play C7 (C dominant 7th) you'd play C, E, G, and Bb. For C diminished seventh, you'd play C, Eb Gb, and A (Bbb). You need to know them well enough to be able to see a chord symbol like the one in the step above and be able to play it without thinking. If you know your major scales, you could have this step mastered in a week.
6. To reward your hard work, pull out the songbook. Find a song you like and play the melody line in the right hand with the appropriate chords in the left, as you're reading them from the chord symbols. You are now playing a song without reading music (in the traditional way, aka Fakebook style). Congratulations!
7. Even though it probably sounds horrible, practice for long enough and you'll sound more and more like what's written there without you even knowing it. You can always go back to the sheet music to see how they're voicing the chords in clever ways that you're not.
8. Next, learn chord inversions: learn to play CM7 like (C, E, G, B), (E, G, B, C), (G, B, C, E) and (B, C, E, G). Learn those four positions for every chord, but only after you're comfortable knowing what every chord is, and have Step Four under your belt. Don't scramble your brain.
9. Learn the pentatonic scale of your favourite key.
10. Add in a couple of notes from it into a song you're comfortable with. Add some more, and take some of the originals out.
11. Now learn the blues scale of that same key and mix the two. By now, you're probably IMPROVISING! Learn those two scales for every key.
12. Look at the chord sequences in the songs you're playing. Try and splice one from one song into another.
13. Learn the 3, 6, 2, 5, 1 progression. Also learn tritone substitutions and the circle of fifths. Play the same songs in different keys.
14. When you're ready, learn chromatic and diatonic harmony. Learn modes and different scales. Listen to different sorts of music from all sorts of time periods, and anything that you can steal harmonic and melodic ideas from. When you've gotten this far, you can easily teach yourself.