Breast cancer is actually a group of diseases. It's abnormal breast cells that can go on to spread to other parts of the body, and can harm a patient.
Breast cancer lumps come in a whole variety. They can be the classic hard pebble-like feeling. They can be sort of rubbery. They generally don't move around through the breast and they're not generally soft, although they can be in some cases.
A lumpectomy is removing a part of the breast that contains an abnormality, usually a breast cancer, but not always. It is not removing the whole breast, but just removing a part of that breast.
There are many different patterns for breast cancer treatment. Most patients require surgery of some sort. Some patients also are benefited by chemotherapy and some are benefited by radiation. So it's often a combination of those three, or two of those three modalities.