Early on in your piano playing journey you learned about rhythm, its the pattens of long and short notes that line up with the beat of the piece. You learned how to read them, and you learned how to play them on the piano. What more could there be to this?
All of the elements of a piece of music change during the piece, that is to say they have their own rhythm. The most important of these is the underlying harmonies, the chords that the piece is made up from. The rate of change of these harmonies is called the harmonic rhythm. Put more basically the more different chords there are in a given amount of time, the higher the harmonic rhythm, and the higher the density of harmonies in the piece.
Harmonic rhythm is an advanced and difficult concept to grasp, and it is something we will work on in our piano lessons and especially when we study improvisation. It is incredibly important. It drastically impacts the mood and feel of a piece, and it has huge implications for the tempo of the piece.
Basically the denser the music the slower it needs to go to make sense to the listener, and so understanding the harmonic rhythm can help find the right speed for the piece. Also denser music often needs clearer and cleaner playing with more control of the sustain pedal, and understanding the harmonic rhythm will hugely help with this.
Like many aspects of music, harmonic rhythm is hard to describe and difficult to comprehend by reading about it. Thats why piano lessons are so important, as a good teacher will open up these concepts for you naturally as you learn to play music.