you are definitely not fat. A healthy weight for you is actually about 120lbs for your height. You should definitely talk to someone about how you feel about your weight because you could be headed towards anorexia or bulimia if you think you're fat at 107lbs.
I'm 5'4" and 107 is a low weight for me. I'm good at about 115. I went down to 106lbs when I was sick in the hospital and couldn't eat and I looked like a skeleton. Right now I weigh 125 and even tho I'd like to lose about 10lbs I still don't think I'm fat.
Please talk to your parents or a guidance councillor or your doctor...talk to someone who's close to you who wont just say "don't be silly". Make them understand that this is a real concern because if you continue to think that 107lbs is fat then you might be headed for trouble.
Given your height and weight I would be inclined to say you're underweight. Though I am a guy, I have done the weight battle and there are just some things that won't change no matter how well you eat and how much you exercise. Also, I am not a doctor, but only having you period once a year would, to me, be a symptom of another issue; given your concern with your weight, I would say malnutrition has something to do with it. Eat right and eat healthy. That is the best advice someone without a Doctorates degree can give you. If you eat too much, you'll gain weight, if you eat too little, you'll set yourself up to balloon in the future as your metabolism will slow as well as cause a number of other issues with your health.
Every girl I have gotten to know since I hit my teenage years has been overly concerned with their weight, and if they weren't concerned with their weight it was something else with their body. Be happy with who you are. Confidence is what you need to like what you see in the mirror.