There are many approaches to sound proofing, many which were already mentioned.
Some few years back, I did a major remodel of an eighty year old farm house. The remodel involved completely rebuilding several exterior walls, installing and removing windows and doors and so on.
The owners gave me a lot of room and I gave them a lot of extras. I got to play and experiment more than a little and learned many interesting things. For example, bored, one day, I grabbed a few tubes of caulking and, with the rock off and no insulation on the kitchen walls, caulked every joint I could find. I caulked 2x's to Tyveck, studs to sill plates, around windows and doors, and so on.
A few days later, I was working and it was really quiet in the kitchen. I walked over to the French door to toss something on to the deck and discovered a significant wind storm was occurring. That was the day I learned how huge an improvement you could make by stopping air flow.
Stopping air movement probably did more than double, offset, no contact walls, double layers of rock and special boards and things.
I already knew sound was rarefaction and compressions of air waves, but, that day, I got to see, in action, what happens to sound when you impede air movement.
Of course, this is not the only solution, but, without it, all the other methods described would be worthless.
Another experiment I did was, in the course of remodeling one of the bathrooms, I stacked layers of rock onto the inside wall of the finished rock. I only went a couple layers deep, but that, combined with an exterior tread plate on the bath door, went a long ways to absorbing sound (e.g., the walls vibrated less, so moved less air on opposite side). Essentially, the bathroom, which was just off a large meeting room, was Taco Bell/Time safe.