My lawn mower will start with a new or freshly charged battery. When you turn it off it will not crank unless you charge the battery again. My question is where is the alternator where I can take it off and exchange it.

You may have a bad alternator but I would check the battery first. I suspect you have a bad or shorted cell. Disconnect the cables and fully charge the battery. Wait a day and see if you still have 12+ vdc. If the voltage has dropped below 12 vdc with no load I would consider replacing the battery. You can check the alternator by placing your test leads on it while running the engine a nearly full throttle. You should read about 13 vdc.

I went and got a good new battery and replaced it and still the same. I cranked it with the new battery ran it a while and turned it off to fill the gas tank. When I went to crank the mower to finish cutting my lawn the battery was dead again. I had to charge the battery to crank the mower to finish my lawn.

You must have a short somewhere on the mower. Disconnect the battery and test the two cable leads for continuity. If there is continunity across the two cables you definitely have a short. I'd start with checking the ignition switch and starter and we'll go from there.