How are modes used when improvising on the piano?

Like the major and minor scales, each mode has its own tonal qualities - in fact, the major and minor scales are themselves modes (ionian and aeolian, respectively). By pairing certain modes to certain chord changes, you can create unique sounds. For instance, the chord progression C major - F major can be paired with a melody based on the major scale, but by using the Mixolydian mode you can make your music more bluesy. A minor chord progression will pair with Dorian mode to create a more modern, jazzy sound.

It really depends on what kind of mood you wish to create, and also what the other musicians you're playing with are doing. In a jazz trio, the pianist will play off of the bassline, but they'll both be paying attention to the chord progressions. There's a lot more freedom there than, say, an 18-piece big band where you're better off limiting your improvisation and letting the ensemble speak for itself.