it's rarely cheap or easy.
sheetrock (wallboard, drywall, it has many names) comes in sound-dampening varieties. some have steel plates sandwiched between the layers to deaden noise, others use different materials. it is expensive, but if you know a builder and don't mind waiting and doing it yourself, you can ask for the smaller offcuts that get thrown away.

otherwise sound studios (or small bands in the garage) on the cheap use dense foam, styrofoam (useless unless layered with other stuff) or the classic, cardboard egg cartons. these dampen sound a little but are obviously ugly - they also eat space like mad, especially if you want your place looking good and then put wallboard over top.

a very different plan: buy a white noise machine. people typically say they sound like waterfalls and such, and can take a few nights to get used to. they do work for most people, though. alternately, buy an old fan that hums or makes a constant rhythmic noise as it spins. you get the noise that replaces the noise of the neighbour and the added benefit of air circulation, too.

gaz me if you have any questions, i don't check in here often.
( postbear has metal knees, Sat 31 May 2008, 7:59, Reply)