The states in which major amounts of gold have been found are: Alabama, Alaska, Arizona, California, Colorado, Georgia, Idaho, Michigan, Montana, Nevada, New Mexico, North Carolina, Oregon, South Carolina, South Dakota, Tennessee, Texas, Utah, Virginia, Washington, Wisconsin, and Wyoming. Although the records are generally not very good, small amounts of gold have also been found in Indiana, Maine, Maryland, Missouri, New Hampshire, Oklahoma, Pennsylvania, Rhode Island, and Vermont -- which taken together with the big producers means that in three out of five states, you have a decent chance to find a little for yourself.
Where in these states should you look? Government records, in the form of geological reports and maps, will tell you where and in what quantity gold has been recovered in the past. These records have been summarized for many states in books such as the Placer Gold Deposits series. And because gold is washed down into placers, areas where a lot of gold has been found by earlier miners -- even if the placer was played out -- will very likely have gold again.
Gravity is the reason gold collects in placers; gold is six to seven times heavier than ordinary sediments. Because it takes a lot more force for a stream of water to move gold, the gold will tend to get caught in cracks and crevices, to settle out where the flow slows, and to work its way to the bottom of deposited sediments. Knowing this makes it easier to figure out where in a stream to pan for gold. Dry streambeds can contain placers of gold as well, laid down by long-gone flows of water. Unless water is nearby, though, you'll need equipment other than a gold pan to recover it, such as a metal detector or drywasher.
One more reason for checking government records before you set out to hunt for gold: some public lands are off limits to prospecting, while other areas have already been claimed by earlier prospectors.
*Much of the information in this article comes from: Gold Mining in the 21st Century, by Dave McCracken; You Can Find Gold with a Metal Detector, by Charles Garrett & Roy Lagal; Dry Washing for Gold, by James Klein.