The invention of the clay brick must be the greatest breakthrough in building techniques. They are so versatile, and good for soundproofing too. But not all bricks are created equal. For good soundproofing, bricks must not have holes in them. Sounds too obvious to say perhaps, but it's true that some bricks do indeed have holes.
Also, some bricks have an indentation called the 'frog', which helps the cement 'key' the wall together. According to BBC recommendations, the frogs should point up, although in common building practice they point down. The advantage here is that if the frogs are uppermost, they have to be filled with cement. The result is a wall that is heavier, and therefore better for soundproofing.
When we move up a size to blocks, well blocks can be made of all sorts of materials. One of the early blocks was the 'cinder block', also known as the 'breeze block'. This is totally useless for soundproofing because as its alternative name suggests, it lets air through, which carries sound. The ideal is a solid, non-porous heavy concrete block. These are excellent for soundproofing, and due to their size the walls go up very quickly.
As always, double-leaf construction is preferred. This means that instead of using two courses of blocks right next to each other, a gap is left in between. Doubling the thickness of a partition improves the soundproofing by 6 dB. However, separating the structures means that not all of the sound that passes through the first leaf actually gets into the second. Some can be absorbed by mineral wool (mineral fiber) in between. In practice, the improvement this brings might be as little as 3 dB or so. But it's free, apart from losing some space, so you might as well have it.