There are two reasons travelers can legally possess a valid second U.S. passport: One is political, the other is logistical.
First, politics. The U.S. government will grant you passport number two if you can prove that you’ll be traveling between two countries that will deny you entry because you’ve visited the other. The State Department’s official language does not pinpoint specific nations, but this currently applies only to the Middle East, where many countries deny entrance to travelers who have an Israeli passport stamp. (Israel now lets travelers request that they don’t get a stamp and instead receive a stamped piece of paper to keep inside the passport; still, the State Department will consider giving you a second passport if you’re visiting this region.)
The logistical reason is a little less straightforward but is one that all serious jet-setters should know. The State Department will OK your second passport if you sent your original away to get a visa for one country but are planning to travel to another in the interim. For example, say you’re going to Belgium, where no visa is necessary, but then plan to go Vietnam, one of 41 countries with a visa program that requires you to send your passport in the mail to one of its consulates or drop it off in person. (Brazil, China, and Russia are the other main tourist destinations on the list.) If your trip to Belgium falls within the window of time you’ll be without your passport—and it can be held up for weeks or even months—you will have no way of passing through EU immigration. A second U.S. passport is the State Department’s work-around.
The application process for a second passport is identical to the original, except you’ll have to prove you fall into one of the aforementioned categories. The second passport comes with its own unique number, is valid for two years (instead of ten), and can be renewed just like your primary passport. It costs the same $170 fee as your first passport. Plan on four to six weeks for the process unless you use a passport concierge to speed things along.