Car not starting after putting in new battery

My car is a 2002 Mitsubishi Galant ES/LS 4 cycl with 93,000 miles. I went to start my car and it would not start. I jumped it and it took some time to get it started with the jumper cables on the battery, but it did start eventually. After I shut the car off and try to restart it, it would not start again. I thought it was a bad battery so I put a brand new one in. It started right up, but later in the day I went to go start it and all I heard was a loud clicking noise. All the lights and electrical system work. The old battery had alot of corrosion on it. I cleaned the battery cables, tips and all very good with a dry toothbrush before putting in the new battery. Please help me to understand what could be the problem.