Dont know much about starters but sounds like the solenoid. i think thats what its called, the round thing sitting on top of the starter. That or a wrong connection
What brand? I just had the same problem with a Hitachi. It was supposedly the right one for my car, but it wouldn't engage. I got a PA from 50resto and put it on Saturday. It works like a charm!
sounds like it aint the right one for your car... I bought a Protorque starter from Summit and it worked perfecly the first time. If what you got now doesnt end up working you should get one. also get some ARP 12 point starter bolts, they work great and dont strip like the stock bolts (not fun removing them at all).