Will a biopsy of my breast determine if it's cancer for sure?

Usually. A breast biopsy, like any biopsy is usually definitive for making a diagnosis. There may be circumstances where not enough tissue is taken (eg, from a fine needle biopsy) and more tissue is needed (core or excisional biopsy). There also may be questions about the diagnosis that can only be resolved by additional biopsies.
Yes. The two most common indications for a breast biopsy are a palpable lump or a mammographic abnormality. Under most circumstances, it is unnecessary to surgically excise these to reach a diagnosis;they can be biopsied via core biopsy using x-ray or ultrasound guidance under local anesthesia. While "false negative" results are very rare, we will typically do a 6-month follow-up study to be safe.
Yes. If a lesion or suspicious finding is found today's core biopsies done by the radiologist or a wire localization excision are state of the art methods to make sure there is no cancer in that area. If this is negative you still need regular exams and repeat mammography in the future as its no guarantee that cancer will not develop later.
Breast biopsy. The purpose of getting a biopsy is to rule in or rule out cancer. Unless the person who performs the biopsy is not satisfied with it then other options are open biopsy to get a large sample or to repeat the biopsy. There always the possibility of false negatives. False positives are very uncommon. See what a general breast surgeon has to say.