For some reason, many people believe that this is true. I believe that it is not.
Technically, yes, if people all decide the currency has value, then it will have value. But this is a ridiculous argument! How much is this currency supposed to be worth? According to this argument, any number is potentially valid! This means that there is absolutely nothing that should hold the value of the currency fixed, and with a few "no free lunch" arguments (anyone can claim any random thing has value and try to trade with it), we see that things with no intrinsic value should NOT have value.
So why does the US Dollar have value? It's just paper (cotton), right? Not true! If the exchange rate of the dollar decreases, you'll see the Federal Reserve will start trading some of its goodies from Fort Knox (gold, etc.) for US Dollars, in order to maintain the dollar's exchange rate/value.
So in reality, the US Dollar is backed by holdings that have intrinsic value.
EDIT: As pointed out, I should mention that another important backing of the USD is taxation.
Also, what I said about the Federal Reserve isn't technically true. What's important is that they have a mission to moderate the rate of inflation of the USD (i.e., maintain its value), and their asset holdings are critical in allowing them to do this.