Despite my greatest efforts, I could not find any information about what standard the U.S. is using at this time. Apparently, we used to be on a gold standard. What about now? Do we even have a standard?
The American dollar is now Fiat Money, unbacked by any physical asset
I decided to answer it even if briefly it is already answered by Semaphore and Pieter Geerkens
The money itself not backed by any physical assets, to understand the current situation, the existing two answers contain the "Fiat money" definition, which is one big part of the story.
The complete story is way bigger than it can be told in this single answer, but I will try to collect all vital, and useful information since I was interested in it for few years now.
First of all, an official source: Modern Money Mechanics from Federal Reserve central bank (mostly privately owned bank), which tells us how money is created out of nothing (actually from some demand of money supply), which means unlike previous money system backed by gold or silver assets (or anything limiting factor) the money supply can be expanded theoretically to infinity (see Hyperinflation), but in reality people lose faith in currency by that time and start bartering.
I would recommend lectures from documentaries like: Money Masters, or Chris Martenson's and Albert Bartlett's lectures available on youtube (the scope of theirs are broader than just money system). Also might be useful Mike Maloney's "Hidden Secrets Of Money", which contains a good part of monetary history.
I wouldn't recommend Zeitgeist movie even if it contains valuable informations, it is sort of manipulative and you might get drifted far from the information which you are looking for.
For summary: the standard of USD technically the faith of the people (not just US citizens, but foreigners too), which is losing rapidly in the past 10 years, for ex China started agreeing not using USD for international trades so far with 8 currencies (most recently EUR), and also there are serious debates on how to trade oil without using USD.