'92 Nissan Sentra possible alternater problems?

Okay so my nissan sentra is a 92 automatic I havee no idea how many miles are on it cus I need to get a spedometer cable but all winter it died and everyone fiured it weas just to cold and the battery to weak and old to start it but now its spring and it kept doing it only the other day it drove fine all day and then it started putting and would nt acceleralrte and it died and so the air filter was dry and there wasnt water in the gas and we jumped it and it started and we let it run for a bit and went to get a drink but then not even ten minutes later it had died. We took the battery home its a newer battery and fully charged it and I got it home the positive side on the battery ther connection thing was loose and the bolts so corrioded I cant tighten it or loosen iit but at least now its firmly in place but this morning I went out to try to start it and it made this really weird sound like really really fast paced clicking almost I guess would be the best wat to say it and then it didnt even try to turn over and now it doesnt even make the noise as I tried to start it. I dont know if its the alternater that bad or how to tell I would like to know for sure before I buy I new one. My engine light nmy check enfgine light and my brake lights are on please help.