You need to know the basic concepts of diatonic functional harmony and chord progressions. There is a Wikipedia article on Diatonic function that will get you started.
The next topic is voice leading and the use of chords in inversions. The concept of voice leading comes from Bach's choral music, but there are rules about voice leading that apply to piano.
Here is a quick and simple example of voice-leading on piano.
In the key of C, the three basic chords are C major, F major, and G major. We call these I, IV and V, because they are the first, fourth and fifth chords derived from the C major scale.
You can play them as in the first phrase, as triads in root position. However, as a pianist, you would want to play them as in the second example: the C chord in root position, the F chord in second inversion (with the C as the bottom note), the C chord in root position, then the G chord in first inversion (with the B as the bottom note) and then back to the C chord in first position. You can play this chord progression without moving your right hand position. As you move through the progression, you'll see that each chord has one common tone with the chord following it (the first through third chords in the progression have the same "C" note, and the third through fifth chords in the progression have the same "G" note in common.) So this is how to play the chord progression smoothly on piano.