During ovary removal surgery, there may be a risk of too much bleeding. This is usually fixed by blood transfusions. There could also be damage to the bladder or bowel. This happens in less than 1% of all cases.
Days or weeks after surgery, you may get an infection. Or you could get a hernia, which is a weakness in the muscle near the cut.
Ovary removal is usually considered safe. Remember that if your doctor advises you to have this surgery, it means the benefits outweigh the risks.