Having the right amount of calcium (and Vitamin D) in the diet is important for keeping bones healthy and strong. Supplements can help make up the difference between what's missing in your diet, but it's considered best to get it through foods rather than supplements. Still, a doctor may recommend calcium supplementation if there is concern that a patient's health is at risk because they aren't getting enough.
Calcium supplements won't cause bone spurs or any other type of calcium deposits in the body. Those deposits are the body's response to inflammation. When something is causing pain or inflammation in the body, it attempts to grow a protective shield to stop the inflammation from doing further harm. Bone spurs are an attempt to grow bone to fuse areas together that are causing problems. Calcium deposits on tendons or other structures are attempting to protect the body as well.
Some literature suggests that getting too much calcium through supplements can cause harm to the body, though, so it's important not to take more than the recommended amount. Current guidelines recommend that healthy adults get about 1,000 mg of calcium and 600 IU of Vitamin D daily, and elderly adults get about 1,200 mg of calcium and 800 IU of vitamin D daily. Again, that's the TOTAL recommended for each day of food and supplements combined. There is usually no need for supplementation if food is providing at least that amount.
If you have bone spurs or other calcifications in your body that are bothering you, or you just have pain in certain areas, an orthopedic doctor can help investigate the cause.