Why does the rear hub (cassette) moves on the axle when I pedal?
It's almost certainly a bent rear axle.  I'm willing to guess that you have a freewheel design, where your rear chainrings have the freewheel mechanism integrated.  The assembly then screws onto the rear wheel.  This design was universally used from the mid-80s back, and is still used on low-end bikes.  The design works ok for single-speed bikes, but as you add more gear sprockets on the back, the drive-side wheel bearings (at the edge of the wheel hub) must be further and further away from the frame.  The resulting forces are notorious for bending axles, even with good quality axles.  I've broken one axle cleanly through because of this issue.  On a low-end department store bike like yours, the axle is probably even weaker, so it's not surprising that it would bend, even if you haven't been abusing the bike badly.

You should be able to replace the rear axle, assuming that there is no additional damage.  It's probably best to take the bike to a bike shop, since the work is a bit messy and is a little finicky with reassembling and adjusting the hub.

If you wanted to do it yourself, you should do a bit of reading on how cup and cone rear hubs work (Sheldon Brown's site has some nice diagrams and advice: http://sheldonbrown.com/cone-adjustment.html and http://sheldonbrown.com/tooltips/hubs.html).  Then you could disassemble the wheel bearings, pull out the axle, find a matching replacement, then reassemble.  You would need wrenches to remove the rear wheel nuts, cone wrenches and grease to reassemble and readjust correctly.