what does Tennessee mean when he yells ''Euchred, old man!''?

"To euchre" is a verb meaning to cheat or swindle. So you might hear someone say that they'd been "euchred" out of several thousand dollars by a con artist, for example. Euchre is also the name of a card game. During a game of Euchre, if your opponent fails to win three tricks after having made the trump, then they are said to be euchred.
As Tennessee is a notorious gambler and card player, this is what he means when he shakes hands with his partner during his trial; he's effectively been euchred, or trumped by the legal system. But then, Tennessee has himself euchred lots of folk in the past, including his own partner. Unfortunately for him, his euchring days are well and truly over. He's euchred one man too many and is now on trial for his life. The clumsy attempt by Tennessee's partner to bribe the judge hearing his case simply makes him all the more determined to sentence Tennessee to death by hanging, which he does.