Well, besides adding personal stress to your life, the lender will report late payments to the credit bureau.
Having a recent history of late mortgage payments may affect your ability to get another mortgage.
You will be charged a late fee - usually 5% of the monthly payment.
Worst case, of course, if you allow late payments to get out of hand, you may face the loss of the property through foreclosure.
Most mortgage payments are due on the first day of the month and have a 15 day grace period. As long as the lender receives your payment by the 15th, you are still on time.
If you are stuck in a situation - temporary - in which you may have trouble making a payment, call your lender and ask if they will accept a payment of interest only or a partial payment. Don't count on them saying yes because it's rare, but some lenders are open to this rather than skipping a full payment.