You'll present your paid homeowner's insurance policy, or a binder and receipt showing that the premium has been paid. The closing agent will then list the money you owe the seller (remainder of down payment, prepaid taxes, etc.), and then the money the seller owes you (unpaid taxes and prepaid rent, if applicable). The seller will provide proof of any inspection, warranties, etc.
Once you're sure you understand all the documentation, you'll sign the mortgage - agreeing that if you don't make payments the lender is entitled to sell your property and apply the sale price against the amount you owe, plus expenses. You'll also sign a mortgage note, promising to repay the loan. The seller will give you the title to the house in the form of a signed deed.
You'll pay the lender's agent all closing costs and, in turn, he or she will provide you with a settlement statement of all the items for which you have paid. The deed and mortgage will then be recorded in the state Registry of Deeds, and you will be a homeowner.