I was reading about the abandonment of the gold standard in favor of flexible exchange rates. Are flexible exchange rates and floating exchange rates the same thing?

When trading nations each employ a genuine gold standard, exchange rates by definition are "fixed."  If for instance, the U.S. defines a "dollar" as one-fourth ounce of gold and France defines its "franc" as one-half ounce of gold, then the exchange rate is automatically 2 to 1 for the same reason that the "exchange rate" of one foot to a yard is 3 to 1.

When the gold standard was abandoned (an unfortunate circumstance, in my opinion, accomplished for the purpose of making it easier for profligate governments to spend and inflate), the result was a regime of paper monies.  "Flexible" exchange rates within that context refers to paper currencies that are free to move up or down in value against one another according to conditions of supply and demand and all the factors that comprise supply and demand.  Because governments and business generally desire more stability in currency values than this situation produces, governments have frequently established some limits to how far their currencies can move in value one way or another before the government will then intervene (by buying or selling its own currency, or other currencies, in the open market).  Allowing currencies to fluctuate but within these designated "bands" is generally referred to as a regime of "floating" exchange rates. 

Both "flexible" and "floating" rates differ from a true "fixed" rate regime in that a currency is allowed to fluctuate.  Some economists and certainly most business reporters in the news media use "flexible" and "floating" synonymously but there is a slight difference between the two, as I've explained above.