A dog's bark can penetrate many types of walls because of its high volume and its wide range of frequency content, which lies in the human hearing range. Different materials absorb and reflect specific bands of frequencies based on their density and elasticity. A typical wall, which is made of wood, drywall or cement blocks, absorbs only a portion of the frequency content produced from a dog barking. Therefore, to soundproof a wall from a dog's bark, layering different materials is required to remove all the frequency bands.
Attach a reflective, high-density material directly to the wall as the first layer of soundproofing. A good material to use for this is half-inch-thick wood or plastic sheeting, and ideally it should cover the entire wall. The first layer should be a high-density component, because high-density materials are generally heavier and thus need to be closer to the wall to be sturdy. These materials are also efficient at absorbing and reflecting high frequencies, such as the high part of a dog's bark or a whistle. Screw each corner of the sheeting into the wall and be sure to use screws that are suitable for your type of wall. If your wall is drywall, use drywall screws. If your wall is cement, you may need to drill pilot holes and use cement screws.
Add a medium-density material on the first layer of soundproofing as the second layer. Cardboard works well for this layer and can be cost effective when gathered from used cardboard boxes. Medium-density material absorbs the midrange frequencies in the human hearing range. Attach material with 1-inch wood nails.
Install a low-density material as the third layer of soundproofing onto the previous layer. To trap low frequencies, use some sort of foam -- the thicker and softer the better. The width is important because low frequencies travel in longer strides than mid- to high-range frequencies, and thus require a larger volume of absorbent to slow its progression. Use wood nails to secure the foam to layer two.
Place a thin layer of cloth over the entire wall as the final layer of soundproofing. Keep a small gap of air between the cloth and the foam to remove the last remnants of low frequency air velocity. Achieve this gap by attaching the cloth with the wall screws an inch or so from the foam. This layer also acts as an aesthetic blind for the other layers, since soundproofing can appear off-putting in a room, depending upon its use.
Soundproofing can be quite flammable. Keep electrical wires and open flames away from your soundproofing, and consider investing in flame-retardant spray and a fire extinguisher.