Engine starter not working? No Crank over?

When I tried to start the car it sounded as if the battery may have been bad. When the car started and I turned off the ignition it restarted right away no problem. I cleaned the battery post and terminals. It started but still sounded as if the battery may be bad. After a few minuets it would not start. I took the battery in for replacement and they told me that the battery was 60 percent charged. I had the battery charged and car still will not start. All the electrical systems seem to be working. Light do not blink when I try to start car. Any suggestions as to what may be the problem.