In addition to the roughly $2,000 you'll need at the time you submit a contract (see Post from 09/18/2010), you should already have money saved IN A BANK ACCOUNT for a down payment. The money that's in your mattress doesn't count because your lender will want to know where your down payment came from in order to make sure it wasn't borrowed from some other lender and will not have to be repaid.
Furthermore, your down payment money should be in the bank for a full SIXTY DAYS before your file reaches underwriting. What this means is that you really should have your down payment money saved up and ready before you start going out to look at houses or VERY soon after you start looking.
So how much do you need for a down payment? Well, that depends on the type of loan you're getting.
If you're eligible for a VA loan, you can buy a house with no downpayment.
For an FHA loan, you'll need 3.5% of the contract price.
For a conventional loan, you'll need 5% of the contract price.
If you have the money saved up, you should put 20% down in order to avoid paying mortgage insurance.