1999 Honda Accord DO I NEED A ALTERNATOR OR A BATTERY

About 9 months ago, while driving about 30 miles per hour, my car turned off on me at the stop light. I started it back up and it turned off again. I did it a second time and I kept pumping the gas at the same time it started and I was fine for about 4 months. The same thing happend again 4 months later more this time more frequently. I would come to a stop sign and my car would just turn off on me. I would start it up and rev on the gas two or three times. Finally the fourth try my car would start and go. Now my car wont even start up. I purchased a new starter with a sylenoid. My brother installed it. We had the same issue. The car wouldnt start. The fly wheel teeth were just fine. I can turn my car on and see the lights light up on my dashboard but I cant get my car to start. The lights in my car are dim. When I try to start it, I hear clicking coming from the starter section. I bought this used car from a private owner 3 years ago. I dont know if she ever replaced the battery or alternator before I purchased the vehicle. I want to say I need an alternator because the alternator takes over while your driving and while I was driving it just turned off on me. However, the battery seems to weak to get the starter going. Iam really confused. Please help if you can. Also how could this have happend over about a 9 month period off and on.