Mechanical part of filtration would be anything used such as filter pads,filter floss, and cartridge 's to remove solid particles from the water.
Chemical would be products that help keep parameters in order like for instance Seachem prime is added to aquarium before tap water is added to remove chlorine which can kill the live bacteria in your filter and on subtrate, ornaments, driftwood etc.
Biological is the most important part of the filter and a must have . Live bacteria is produced by the nitrifying cycle where they grow on bio-media such as sponges, biomax, lava rocks,plastic pot scrubbers, bio balls just to name a few. The live bacteria convert ammonia into nitrite and nitrite is then converted into nitrates. The bio-filtration must be enough to support the bio-load of the fish in your aquarium if not enough it will not sustain life in the aquarium properly.