Without getting too technical, taxpayers with children that had adjustable gross income (the number at the bottom of the first page of the 1040) of between approximately $30,000 to $120,000, and have filed their 2002 tax return, will receive payment of $400 per child.
The $400 represents the difference between the new (accelerated) child tax credit of $1000 per child this year, vs. the $600 that was originally to be paid this year, before Pres. Bush signed the new tax bill.
I know I'm missing a few key points, but it's early, and I don't feel like looking up everything. You can find all of the info at the IRS web page.