In a bi-weekly payment, how is the money applied if the bank only has half a payment?

The first half-payment isn't applied to the mortgage. It is placed in a holding account until your second half-payment is received. Then this full payment is applied to the mortgage on the due date. As you continue making your bi-weekly payments there will eventually be a surplus of funds in the holding account which will be applied as an additional principal payment which will shorten the term of the loan.

I'm guessing you have set this up with your current lender? You can't just start paying bi weekly on your own. You will need to either set it up with your current lender or through a third party provider. They will give you all the information needed to successfully start a bi weekly program, and once set up, no you won't be late on your payments.

You have to set up bi-weekly payments with the lender. You can't just send in 1/2. Most lenders charge a one time set up fee. While it sounds like a rip off, if you calculate your long-term savings - it is well worth the price. By making bi-weekly payments, the average 30-yr loan will pay off in about 23-years.

Charlie gave a perfect answer.

Charlie has it right, but be sure to start with an extra half payment at the middle of the month, so the lender does not charge you a late penalty every month thereafter.

You can also pay extra to your payment with your monthly payment to enjoy very similar benefits without changing your payment schedule.