The bench press is a compound exercise that is performed with a barbell or dumbells and weight bench. During this exercise, you lift the wieght to and from the chest from a face-up position. Being that this is a compound exercise, it works multiple muscle groups at once.
The anatomical name for the chest muscles is the pectorals. The pectoralis major is the part of the chest that gets activated when you are pushing the bar straight up. The pectoralis minor gets recruited when your shoulders shrug forward. Of the two muscle parts, the main emphasis is placed on the pectoralis major.
By changing the angle of your bench press to a declined or inclined position, you would place more emphasis on the lower chest and upper chest respectively.
The shoulder muscles get worked any time there is elevation of the upper arms. Although you are in a lying position while doing a bench press, your upper arms are perpendicular to your body. This causes your shoulders to get activated. They actually get called into play the whole time the bar is moving up and down. The anatomical names for the shoulders are the deltoids and trapezius. The "delts" are on the sides of the shoulders and the "traps" are on the top. Most of the emphasis gets placed on the anterior, or front, delts.
The triceps are the major muscles on the backs of the upper arms. The main responsibility of the triceps is to extend the elbow. This takes place when you push the bar off your chest and upward, moving your arm from a bent to straight position.
Although not commonly thought of with bench presses, the abs are a very important muscle group for proper execution. When you do the bench press, you have to forcefully engage your abs to generate power and to keep your body stable.
The biceps brachii is the muscle on the front part of the upper arm. When you are doing a bench press, this muscle gets used as a stabilizer. It helps keep the arm from moving back and forth during the lift.