This is a very wide and controversial topic. The main reason that countries use the gold standard is to prop up their currency so that it is secured by something material as opposed to totally unsecured. This is because, theoretically, Fiat money (paper currency) could become worth very little due to inflation (see WWII Germany). Pegging your currency to gold prevents such massive value fluctuations.
Many see it almost as a moral issue, that government either shouldn’t or is incapable of effectively managing a nation’s currency. A corollary of this is that a gold standard would eliminate a lot of volatility of the market as investors no longer would have to guess Fed moves or lending policy, as the value of the currency would be in a stable (until recently) commodity.
Probably the main benefit would be a reduction in volatility of a currency’s value, but it must be warned that the effects of such a move by a country as large as the United States would most likely result in a lot of unforseen consequences, both good and bad.