Starter clicks, but doesnt engage

Thanks in advance for looking at this. My starter clicks when I turn the ignition, and my battery is at 12.3 volts. I replaced the solenoid and cleaned the connections to the solenoid, but this hasn't changed the situation. So, like i said, the car is in tip top shape except that when I turn the ignition the starter clicks but nothing else happens (engine doesnt turn over). I'm inturpretting this as the starter not engaging for some reason. Any ideas?

Remove the positive battery cable at the battery and check both wires for corrosion within the connector(known problem) and if good than replace the starter motor.
Post back with the findings.

cleaned the battery cables and got her to start. However, now my car will shut off abruptly when I run too many electronics, namely the lights. I just installed a new alternator, so other than that any ideas as to what it could be?

Did the alternator change correct the problem? and have you had the battery load tested?

the alternator did improve the problem, but didnt completely cure it. I went back to checker auto and was very mad to hear that they never properly tested the alternator, they just assumed and then told me thats what it was for sure. Well, I went back and they had to admit it might not have been the alternator, but of course I couldn't get a refund. After the new alternator was in and the problems presisted, (though not with as much frequency) I recleaned the battery terminals so that they were immaculate. Since then (its only been a day and about 50 miles) I havent had a problem. So, who knows, maybe thats all it was. BTW, where exactly does the ground wire from the battery go to? I'm sure I looked right at it, but I could not identify it to see if it was making proper contact.