A biopsy is the only way to truly verify if you have cancer. A small set of cells are taken out from the area of interest in the breast (usually using a needle) and examined under a microscope. These cells are then examined under a microscope to see if they are cancer cells.
Q: If biopsies are so accurate, why aren't they performed more often than mammograms or breast MRI's?
A: There are two reasons. First, biopsies take a very tiny amount of cells from the tissue, so in order for it to detect cancer, it is important that the cells taken must be from the area with cancer. In order to find that place, you must be scanned first. Second, biopsies are a more invasive and painful procedure than simply scanning the body, which is painless. It also requires more time on the physician's part to examine the cells, so it is performed only when there is enough suspicion that something is there.
If you feel that you may have cancer but your doctor doesn't think you need a biopsy, follow your instincts: talk to your doctor or get a second opinion.