A common do it yourself project in home brewing is converting a cooler into a mash tun. When building a mash tun, there are several options available to use for the filtration system. It will likely come down to personal preference, but it will also depend upon what type of sparging you use. The basic options are a mesh filter, also known as a bazooka filter, a manifold, or a false bottom.
If you will fly sparge, then you will want to use a false bottom. The bazooka filter is simply a piece of water supply line, such as a toilet supply line hose that comes with a stainless steel mesh cover. Carefully remove the outer mesh cover from the inner vinyl hose, and discard the hose. You then connect the mesh cover to the spigot with clamps, and this acts as a filter for the wort. The wort passes through the mesh, but the debris from the grain gets left behind.
This is probably the most popular design for home made mash tuns in the home brewing world. Alternately, some brewers choose to build a manifold using pipes made from copper or CVPC. The pipes are connected in a square or rectangle, or as arms branching off. This manifold attaches to the spigot. Tiny holes or slits are drilled or cut into the pipes to act as filters. If you choose to use a manifold system, be sure you do not glue the pieces together. You will want to be able to take the apparatus apart after each use to clean and sanitize.
False bottoms are another option, and are usually the choice if you will fly sparge. With fly sparging, you are constantly pouring water back over the grain bed. You want to avoid channeling, in which the water pours too heavily in certain parts of the grain bed, creating channels. This only rinses parts of the grain bed, while leaving other parts unrinsed. A false bottom prevents this, and allows the wort to flow evenly through the grain bed to ensure all the grains get thoroughly rinsed and the sugars are fully extracted.
Another option is one that is closely tied into the BIAB home brewing method in which no filter system is used. Instead, you use a standard cooler that has not been converted. You place a grain bag into the cooler, stretching the bag over the sides of the cooler. Then, pour the water over the grains just like you would for a batch sparge.
This accomplishes the same result as a batch sparge, but does not require a converted cooler. This may be an option if you are new to the world of all grain home brewing, but converting a cooler will probably be worth the effort as it would increase the efficiency. Plus, this step is more for sparging. You would not be able to mash in an unconverted cooler, but if you want to try the BIAB method before jumping in and converting a cooler, you could try this.