There are a few different possible causes of this type of pain. Right sided pain can be caused by problems with the gall bladder, pancreas, liver, appendix, and sometimes the intestine. If your pain is on the upper right side, then the gall bladder or pancreas are the most likely sources (from a gall stone or pancreatitis). If your pain is in the lower right side it could be appendicitis, or perhaps some inflammation of your colon.
The fact that you feel like you are going to lose urine when you press on your stomach suggests to me that you may have a very full bladder. If this is the case, your may be retaining urine in your bladder and having trouble getting it out. The most common cause of this in men is an enlarged prostate gland. In women, the most common cause is a medication.
Since your description does not narrow the possibilities down very much, I would have to suggest that you see your primary care physician
for this problem. He or she will perform a thorough abdominal exam and figure out exactly where your pain is coming from. If your pain is coming from your bladder area, then your doctor
will perform an ultrasound
to see how much urine is trapped down there.