In accounting, the word "balance" means the current value of an account after all the pluses and minuses are applied. An account could be a bank account, a credit card, the equipment owned by a company, or anything else that we want to attach a monetary value to. Many accounts have money constantly going in and out. Your example appears to be about some kind of medical bills. So every time you visit the doctor (or whatever), the amount you owe goes up. Every time you make a payment, the amount you owe goes down. For some types of accounts the balance could be either positive or negative. Like if you owe $100 but for some reason you pay $110, the amount you owe is now negative $10.
When you owe money, there is usually a "due date", the date that you are expected to get it paid. You may have some amounts on an account that are now due and others that are not. In this case, if you have a charge, then they send out the bill, then you have another charge, the first charge is now due but the second charge may not be due until next month (depending on how they do their billing). With medical bills, often you don't owe anything for a visit until after the insurance company pays whatever they are going to pay, and then you owe whatever is left. So the "balance due" is the amount that is due now. It may not be the same as the current balance if you have other charges that are not yet due.