You have a problem with your starter solenoid. You either need a new one or the contacts are dirty. First try tapping on the solenoid with a screwdriver handle while somebody turns the key. If that doesn't work, try wiggling the cables at the solenoid, then try starting it. If that doesn't work, clean the connections with a wire brush. If that doesn't work, try tapping on the starter with a hammer while somebody turns the key. If the last thing works, you need a new starter motor. If the first thing works, you need a new solenoid unless your contacts were loose or dirty.
Bruce