What card game do the characters play in "Of Mice and Men"?

Just to add a bit to Mrerick's answer:

Euchre is what is known as a "trick card game." It's simple structure belies a complexity that involves psychological bluffing as well as sometimes intricate mathematics.

For Steinbeck to have chosen the ranch hands to play this game is symbolic of the intelligence of his characters, who though poor and uneducated are themselves more complex than their "simple structure" might appear.

What card game do the characters play in "Of Mice and Men"?

Euchre.

The suggestion is made as a distraction from thinking about Carlson about to shoot Candy's dog.

What occurs during the card game in "Of Mice and Men", and how do characters feel?

All quotes used are taken from the Penguin version of "Of Mice and Men" (1978).

The men begin the card game because they are trying to distract themselves while Carlson takes Candy's old dog outside to shoot it: "The way I'd shoot him, he wouldn't feel nothing." (Pg. 45) George does not shuffle the cards: "He rippled the edge of the deck nervously..." (Pg. 49). Candy is laying on his bunk staring at the ceiling. The men know how much that dog meant to Candy, and they are nervous, sad, and feeling sympathetic about Candy losing his companion. They are all waiting in anticipation for the sound of Carlson's shot coming from outside, and they know Candy is too. Everyone is anxious for this difficult scene to be over with.