Prorated rent is the amount you pay for living in a rental for only part of the month. It's calculated relatively in accordance with how many days out of the month you live in the rental.
If you move in halfway through the month, you don't want to be stuck paying for a full month's rent. Instead, you can pay a prorated rent price for only the last half of the month.
Landlords aren't always required to prorate rent or may not do so unless you ask them.
You can calculate prorated rent by dividing your rent by the number of days in the month. Then, multiply that number by how many days in that month you'll be living in the property. This will give you how much you owe for rent.
Monthly Rent / Days in Month = Daily Rent
Daily Rent x Days in Month You Are Paying For = Prorated Rent
There aren't many laws that require a landlord to prorate rent, so understand that you may have a hard time convincing them to do it for you. If you've signed a lease and there isn't a specific term about prorated rent, you can't really argue it.
So, if you decide to move out before the month or your lease is over, you can always ask your landlord to prorate the rent, but they may or may not choose to do so.
But if you're supposed to move in and your rental isn't ready for you until a week later, it would be appropriate for the landlord to prorate your rent since the move-in day was specified in your lease agreement and your landlord didn't comply. In this case, there may be laws in place that would require rent to be prorated, but make sure you check on your state's laws concerning rent proration.