This two-player game is played with a 32-card French suited pack, the cards of each suit ranking from high to low K-Q-J-A-10-9-8-7. Five cards are dealt to each player, one is turned to determine the trump suit, and the object is to win at least three out of the five tricks. The name of the game presumably comes from the fact that if neither player will undertake to try to win three tricks with the hands they are dealt, both may discard unwanted cards from their hands and are dealt an equal number of fresh cards in exchange.
Écarté was popular as a gambling game in 19th century Paris. It survived in Germany until the 1960's, and at the start of the 21st it century is still played in France in some 'cercles de jeu'.
Further details can be found in the entry for Ecarte in the 1911 edition of the Encyclopedia Britannica.
Roland Scheicher has written a detailed set of rules of Écarté in German. A shortened version of this is available on the German Wikipedia site.