I'd like to sound proof my basement as I live in a bungalow and the music is quite loud in the bedrooms above.

I'd like to listen all night and not disturb people upstairs. How can I do this?
To really soundproof a room is difficult and expensive. Most people get by with dampening the sound using insulation between the floor above and the basements ceiling. This can be pretty effective, but you will still have sound/vibration transmitted through the floor joists and the up-stairs walls.

Have you tried a good set of headphones?
We recently wrote an article on sound proofing. It's on our resource page. In the column titled "Rives Articles". It should answer some, perhaps all, of your questions.
I'd like to do this economically. My plan is to build a room within a room. I want to fill the ceiling joists with acoustic insulation available at Home Depot and then drywall over top (2 ply if needed.)

Then I will put in a drop ceiling leaving a gap between. For the side walls I plan to use the same type of insulation then drywall overtop then leave a gap and put in another drywall all the way around with staggered studs.

I realize this won't be perfect but do you think I can get about a 50% reduction in noise upstairs? If so then I'm happy.

Should this plan be changed somehow in order to obtain my 50% noise reduction goal?
Recommend using Roxul fireproof batts between ceiling joints and wall studs.
Kel--A room within a room is the ultimate in sound isolation. When building it's not that much more expensive--but you eliminate structure born vibration. The only way to step it up is to consider the floor (putting a sound absorbing matt under the subfloor), the HVAC--this can be a major sound leakage and is discussed in the paper. Doing a room within a room will not be effective if you have major sound leaks like the HVAC or recessed cans (we see that mistake often).
We recently moved into a semi-detached and my better-half is disturbed by airborne and impact noise from next door. Long story short, we've had an engineer, an architect and a sound-proofing specialist inspect our unit and make recommendations.

Kel, based on my understanding of what has been recommended to us, it sounds like your plan could result in at least a 50% reduction. Your approach should address both airborne (voice, music from speakers, etc.) and structural (impact) noise to some degree. We are proceeding in a first step with blown cellulose insulation to fill the gap between the drywall and the cinder block dividing wall on our second floor and in the attic. The sound-proofing specialist expects this to yield a 50-60% drop in airborne noise.

If that is sufficient for my wife to be able to sleep without being woken, we will probably stop there. If not, the next step will be a false wall. The key to the false wall, the second layer of drywall as you have described it in your plan, is to ensure that the studs do not come into contact with the first layer of drywall. Leaving a 1" gap between the first drywall and the studs for the second drywall is ideal.

Best of luck to you.

Remember one important thing: THE AIR VENT.

I thought I'd soundproofed my basement ceiling with unfaced fiberglass insulation, then 1/2 inch styrofoam panels, then a suspended ceiling with sound retardent ceiling panels.

Guess what? The sound flowed right into the air duct and up into the first floor.
Thanks for the encourgement Max.

Krell man how did you fix your air vent problem and what was the end result like when it was all finished?
Good place to start your search is here:
Hey man, I relate to you, I live beside a highway and it gets really loud especially at peak hours with the honking. Even at night, the engines from vehicles passing by is loud. My neighbours recommended this noise barrier wall for residential home, it worked great and I can catch good night's sleep, I'm sure it will help your case too.
Another important consideration is to make sure any air gaps are sealed with acoustic caulk when framing for example.  HVAC noise can be reduced by inserting a "box" in the line that is lined with 703 insulation.  These techniques and more worked when we were building my brother's recording studio.