When you hit your elbow, or any other part of your body really hard, there are a couple of possible explanations for the pain and soreness that follows.
One is that you actually broke a bone. This is pretty unusual after the type of injury
you describe, as the elbow is a strong bone. Also a broken bone usually has pretty severe pain and there may also be trouble bending the elbow. Nevertheless, if you are concerned or if you have any risk factors that raise your chances of having fragile bones (such as osteoporosis) then it would be worth going to see your doctor
to have an x-ray to rule out a broken bone.
More likely is, as you describe, a "bruise" or "contusion" which is the medical term for an area of soft tissue (skin, fat, muscle) that becomes inflamed and sore after being hit. A 'bone bruise" would be a contusion that affects the lining of tissue that covers the bone, called the periosteum. In effect, however, it is usually difficult to tell the difference between a "bone bruise" and a "muscle bruise" as both cause soreness and swelling at the site of the injury. Both are managed with pain killers and rest.
Talk to your primary care doctor
for more information about this issue!