What is the ammount a parent can give to a child without being required to pay gift taxes?

You can give $13,000 in 2009 (the number changes most years) without having to report it or file a gift tax return. If you give more than that, you have to file a gift tax return. The excess over $13,000 is subtracted from both your lifetime gift tax allowance and from your estate tax allowance. Once your $1 million lifetime gift tax allowance is used up, you have to start paying gift taxes. Once your estate tax allowance is used up, after your death your estate will have to pay estate taxes.

Note that tuition paid directly to an educational institution or medical bills paid directly to a medical services provider on behalf of your child (or anyone else) do not count. You can pay as much of either of those as you want and it will not count against your annual or lifetime limits.