There are no days of "free" mortgage time, you are always charged interest the day your mortgage funds. Mortgage interest is typically paid in arrears, meaning the payment that is due on March 1st has two parts - the principal portion that is due for March & the interest that accrued during February (28 days).
If you close on January 20th, then on the closing statement ("Final HUD-1" it is called), you will be charged interest by the lender from January 20th until your 1st mortgage payment starts covering the interest - meaning January 20th through January 31st (12 days of interest). Paying that interest at closing is called "pre-paid interest", because you are prepaying the interest rather than paying it in arrears.
If you were to close on February 3rd, and asked for a 1st mortgage payment due date of March 1st, then instead of being charged interest at closing, you would be given an "interest credit" (instead of being charged interest) from the lender of 2 days (Feb 1st & Feb 2nd)... and your March 1st payment would be a normal full payment.
January 20th 2013 is a Sunday though, so there isn't any way for you to close then. January 21st is when you would actually close, so change the above examples accordingly.
Shane Milne | Lending in all 50 states | NMLS #81195
Followup and correction from previously answered question. Mortgage interest is paid in arrears. So as Carl says below, you will be paying 11 days interest in escrow at closing. you will see in on the closing statement. Again, your 1st payment will be due on March 1st. this will be the February interest and a small amount of principle. Lender's set up the 1st payment on the 1st of the month and every month thereafter. I hope your home turns out to be a blessing.
Robert McGuire ASR
Your Castle Real Estate
Direct - 303-669-1246