Some of these pictures and descriptions may give away plot details that you might not want to know before watching the film.
Before he goes to Iraq, future war hero Chris Kyle, played by Bradley Cooper, is shown at a shooting range as part of his training to be a Navy Seal sniper. The sniper training sequence is intercut with scenes of Kyle courting his future wife, played by Sienna Miller. At the range we see him shoot too low and miss his target twice. His instructor berates him. Then we see a love scene where he carries his future wife into the bedroom in her underwear. Then the instructor asks him if his off-eye is shut. Kyle tells him if his off-eye is shut he can't see what's out there and the instructor yells "there's nothing out there but the target." Kyle says "negative sir, there's something out there" and the instructor gets mad and tells him to "punch it" then threatens to make him do 50 push-ups. Kyle moves his rifle sight to the left of the target where we can just barely see a snake coiled up in the weeds. Kyle shoots, the snake flies up in the air in a cloud of dust and lands, and we see the instructor with a slightly surprised look on his face. Kyle then says "I'm better when it's breathing" and the instructor looks down with what appears to be a look of approval.
There's no way tell what kind of snake was used, because it's distant and out of focus, but it's almost certainly supposed to be a rattlesnake and its shape confirms that. The credits list a Snake Wrangler, which indicates that a live snake was used instead of a dead snake or a fake snake or a visual effect, and the lack of a no-animals-harmed disclaimer leaves open the possibility that one or more snakes was actually harmed or killed. I suspect that the snake was rigged with some kind of tiny wires to pull it into the air when it was shot. Making the snake fly into the air was probably necessary for us to believe that the instructor could see what Kyle shot.