Well they are 3 different things. For your tank, mechanical filtration would be the pre filter sponge, chemical filtration would be the activated carbon in the filter and biological filtration is your live rock/sand.

Mechanical filtration generally removes larger particles from the water.

Chemical filtration somehow cleans the water through a chemical reaction in some sort of filter media i.e. activated carbon removing the yellow tinge of water or granular ferric oxide removing phosphates in the water.

Finally and probably most importantly is biological filtration. These are the nitrifying bacteria that make your tank go through nitrogen fixation (or the nitrogen cycle which is basically cycling your tank) and turn Ammonia excreted by fish and other inhabitants into nitrate. This is constantly happening. You remove excess nitrates with partial weekly. water changes.