Only your hand, foot and body.
When bouldering, you are generally climbing close to the ground, so you don't need to rope up.
You need only
Climbing shoes, chalk bag and clothing.
Climbing shoes - The most important factor is finding a shoe that fits your foot well.
Chalk bag - A chalk bag holds gymnastic chalk so you can dry the sweat off your hand.
Clothing - you want clothes that move with your body and do not bind.