On a piano what does the left-hand part play, is it the counterpoint that is being played with the left hand? While the right-hand plays the melody?
Creating melodies on piano seems easy, but how do people create the left-hand part that makes the music sound much better like for example here at Badinerie - J.S. Bach BWV 1067.
As Tim already mentioned in the comments, it could generally be called accompaniment.
You can categorize these in different types like:
Each of these can be categorized further like for example a baseline could be:
I think how these are typically build, depends a lot on the genre of the music. But they will all have in common, that they add some harmonic and rhythmic context to the melody.
Generally speaking, you could try to analyze the underlying harmony of your song/melody and use these harmonic information to build one of the types of accompaniment I have listed above. Of course some theory knowledge will help you to analyze the harmonic information and choose how to make use of it.
Your question seems specifically pointed towards music in the style of Bach. I'm not very experienced in this type of music, so I will let this open to other users which might have a better grasp of explaining how an accompaniment could be build in this style of music.
As for the question if the left hand is counterpoint, Wikipedia says
In music, counterpoint is the relationship between two or more musical lines (or voices) which are harmonically interdependent yet independent in rhythm and melodic contour
So one could say, that you could play musical lines with your left hand which will create a counterpoint with your melody in the right hand, but your left hand playing will not necessarily be counterpoint.
you have to look up under accompaniment styles:
google search accompaniment styles
The software Band-in-a-Box offers hundreds of styles which include the rhythm section too - of course.
https://www.pgmusic.com/tutorial_realtrackslist.htm
In this song I have written the melody and the counterpoint, all the rest is Band in a box accompaniment (except the brass fill ins in the chorus and at the end:
Creating melodies on piano seems easy, but how do people create the left-hand part that makes the music sound much better?
Well, this part can't be answered quite easy. You need some experience of listening and composing melodies and counterpoint to write in the style of Bach or Mozart. As you see in my own example I use in my counterpoint imitation, opposite movement, triads. Thus it is the experience of listening and playing, but also the study of chords and harmony that enables you to write an accompaniment for the left hand (or an entire ensemble). Bach has used also triads and imitation and other elements I have mentioned. The hints that Olli has given are all useful. The best way is listening, playing and analyzing lot of music and experimenting with these elements like you were playing with lego stones:
You need to have the music in your mind (and heart) that you can build your world like these lego-domino cracks:
You see: you need a lot of studying and practice and preparation to let the music flow like Bach, Mozart (and many others ...)
