Listen to a wide variety of jazz, from it's beginning through the present day. Seek out recordings with renowned jazz pianists. Since jazz in much about mood and feeling, and is mostly improvised, you must connect with the music on a metaphysical level. On a practical level, the more you listen, the more your ears will learn what 'sounds' like jazz.
Learn all your seventh chords, their inversions, and their various voicings. Seventh chords are more prevalent in jazz than in pop, rock, or classical music, and have much to do with the 'jazz sound'. An inverted seventh chord, is when you position the notes in a different arrangement than in root position. Seventh chords have three types of inversions: first, second and third inversion. Use inversions to smoothly move from one chord to another, or to add color. Voicing refers to the notes that are chosen to 'voice' the chord when playing one chord with two hands.
Familiarize yourself with the jazz standards by practicing with "The Real Book". "The Real Book" comes in different volumes, and is a collection of the 'standard' jazz repertoire. Start by playing the melody with the right hand. Play as many songs as you can in this fashion, as a part of your daily practice routine. When comfortable with a few melodies, play them along with a metronome, starting with a slower setting. Practice playing the chord changes with the left hand, in time, in root position. Once comfortable, play both hands together–the left hand playing the chords, and the right hand playing the melody.
Practice scales and arpeggios on a daily basis. Play them with a metronome using various tempos, to improve your feel for time and your overall dexterity. When working with "The Real Book," play the arpeggios of each chord in the song. This will help your ear to prepare for improvising over the changes.
Acquire a working knowledge of jazz music theory. Music theory is essential to improvisation and understanding how jazz works. Playing the piano helps immensely in understanding music theory, and there are many user-friendly books and websites available. You can also find a class or a teacher, but be sure they are teaching jazz theory, because classical music theory is much different.
Practice improvisation. The real magic of jazz happens when a group of musicians improvise together and off of one another. This is ultimately the best way to practice improvising and playing jazz. In the beginning stages, though, there are other methods for practicing improvisation. Play the chords, or the chords voicing in the left hand, while improvising with the right. Keep the improvisations simple at first, maybe just playing around with the notes of the chords. Try to really 'hear' what you are playing–don't play notes just for the sake of playing notes.