Budh or Mercury Puja is done to appease the planet Mercury (Budh). Planet Mercury is considered to be associated with knowledge and wisdom and so to appease him.
Those who are weak at studies, experience lack of concentration, have skin related health issues, problem related to nervous system etc., should worship Budh. Also those running Mercury (budha) dasha or Sub-period.
A prayer to Budh, on Wednesdays, brings in manifold benefits like removing obstacles, possession of knowledge, concentration, wealth, lands, etc. The Budh Puja is done with faith to ward off all evils of planet Budh (Mercury) as per the horoscope.