I'm just wondering, if you have symptoms of an ovarian cyst, does that mean it is a cancerous cyst?

No: Most ovarian cysts are benign and you needn't worry. Most ovarian cysts are small and are only felt intermittantly as pain. At your age, the chances of cancer are very small unless you have a strong family history of ovarian and/or breast cancer. Ultrasound are good at detecting the type of cyst you have, so ask your doctor for one if you're worried.