Filtration is a method for fining mead. Other methods can be seen at the fining agents page. This is a top-tier approach to fining which includes more equipment and process to accomplish. It is by no means necessary and excellent results can be had via commercially available fining agents.
Only certain setups and filters are effective for filtering mead. Coffee filters and cheese clothes (or similar) are not. They will not reduce lees and will rapidly oxidize your mead.
Filtration units have pore sizes measured in microns, which indicates the maximum particle size that will make it through the filter. Filters are also classified as either absolute or nominal. Absolute filters will filter all particles larger than its micron rating.
Nominal filters will filter most particles larger than its rated micron size, but some will get through.
The wine filter most commonly used by home brewers is the Buon Vino Minijet (Review by u/phiIIips). This plate and frame style filter offers three grades of filters. If your mead is cloudy, you will want to start with the #1 filter, then follow up with filtering with the next grade. If your mead is mostly clear already, you can start with the #2 filter. Most brewers stop at #2. IMPORTANT: they market their #3 pads as 'sterile', however they are nominal and not absolute filters so they are not capable of stabilizing your mead and preventing refermentation after back sweetening.
It is possible to match (or exceed) performance of readily available models. The setup will need 2 vessels (pre- and post-filter), the filter, and a method to move the mead. Moving will involve either a pump or pressurized gas. Gravity provides very little pressure compared to the resistance of the filter.
The simplest example is a pressurized keg-to-keg transfer with inline filter unit. Anyone who has a kegging setup already has 3/4 pieces of equipment. It requires 2 kegs, a CO2 tank (plus regulator), and filter.
Common filter units are 10 inch canister filters or plate filters. Canister filters are often sold for house water filtration and readily usable in mead filtration. Plate filters can be purchased from brew supply vendors. Both have the same function.
Plate filters have less volume inside their housing. It is recommended to purge canister filters of oxygen (directions below). Filters are recommended to be thoroughly 'rinsed' before use. Flow enough water (typically 5 gallons is enough) to accomplish this. See example protocol for directions)
This will be a 2-keg CO2 transfer through a 1um filter. The source keg has the mead and the receiving and filters will be prepped to be used. The receiving will be purged of oxygen and the filter will be rinsed at the same time.