You stand in front of a special x-ray machine. The person who takes the x-rays (always a woman) places your breasts (one at a time) between two plastic plates. The plates compress your breast and make it flat. You will feel pressure on your breast for a few seconds. It may cause you some discomfort, feeling like squeezing or pinching. But, the flatter your breasts during the procedure, the better the picture.
Usually, two pictures are taken of each breast -- one from the side and one from above. The whole procedure takes only a few minutes.