Generally the U.S. has run perennial trade deficits with the rest of the world because people around the world have been essentially collecting dollars instead of spending them. They were for a long time considered the main "hard currency" other than volatile and politically-sensitive gold.
In other words, we would send dollars to country X and receive goods in return. Instead of sending back those dollars to us and receiving goods from us, they would stuff them under a mattress, or in their government reserve vaults, because their own local currency was not considered as trustworthy.
In recent years, foreign governments have begun viewing the euro as a strong, stable currency, and have been diversifying their reserves by getting rid of dollars and stocking up on euros. Two of the recent effects of that phenomenon have been a declining dollar (as with anything, its price goes down when there is a sudden flood of sellers), and high U.S. asset prices (because now the dollars are coming out of the mattresses and vaults, and being used to buy stocks and property).
Instead of sending back those dollars to us and receiving goods from us, they would stuff them under a mattress
Usually they take the cash and invest it in US Government Bonds or private stocks and bonds. This high demand for bonds means our government can offer them at really low rates, and there are still takers. (China has something like $300 billion of US Bonds.) This is an added bonus for us. In this way, our currency's primary status is kind of self-perpetuating. Our currency is backed in faith in the solvency of the US Government, and our Government can borrow money really really cheaply to deal with any crisis it needs to, keeping it solvent where other governments would fail.
(For anyone out there that is scared what will happen if the rest of the world started dumping it's dollar-back assets, remember that less than half of the 'National Debt' is owed to foreigners, and many of these countries use dollars to stabilize their own currencies, so this would amount to financial suicide for them.)
In other words, we would send dollars to country X and receive goods in return. Instead of sending back those dollars to us and receiving goods from us, they would stuff them under a mattress, or in their government reserve vaults, because their own local currency was not considered as trustworthy.
In recent years, foreign governments have begun viewing the euro as a strong, stable currency, and have been diversifying their reserves by getting rid of dollars and stocking up on euros. Two of the recent effects of that phenomenon have been a declining dollar (as with anything, its price goes down when there is a sudden flood of sellers), and high U.S. asset prices (because now the dollars are coming out of the mattresses and vaults, and being used to buy stocks and property).