sknhgy 

The bike is a Trek 820 mtb. I noticed the rear wheel had some play in it. You could "rock" it when it was on the stand. There was some looseness in the bearings. It felt the same as a loose wheel bearing on a car. If you grasped the top and bottom of the wheel you could rock it back and forth and almost feel a clicking.
I have the freewheel removal tool, so I removed the freewheel. There is a hex nut on each end of the axel. I removed the one on the side of the gears. Underneath there was a cylindrical nut with flats on it. It was loose. By loosening and thightening this nut it seems that you can adjust the load on the bearings (am I correct so far?).
It seems like the nuts that hold the load on the bearings have come loose. If that is true, how do I fix it?

Do I tighten the cylindrical nut, then lock it in place with the hex nut? If so, how much do I thghten the cylindrical nut. How much pre-load do you give it?

Thanks in advance. I have included a picture. The cylindrical nut goes on the threaded shaft.