I got summons to go to court the 20th of this month for my 9month old son but his father said he is going to drop it. If he drops it will I still have to attend court and if not can I leave the states to go visit my mother?

Assume nothing. Sometimes a person will tell that lie hoping you don't show up... and when they do they win by default. I would only believe the word of a court clerk that a case has been cancelled.

You need proof in writing directly from the court that it is dropped. How do you know it isn't a trick so that he will win by default (by you not being there). Don't make plans for after the 20th now.
Ask again when all is settled. I wouldn't leave the US any time soon.

Sniped by smoothy!