Smelly fart is a derogatory term used to describe a unpleasant and repugnant gaseous emission produced by Humans and animals. The term is also used to refer to a unpleasant person in a derogatory sense. 
Farts, properly and scientifically known as flatulence ,  are described as "smelly farts" when they have the additional characteristic of having a particularly repugnant odor . Odors such as those of a smelly fart are considered by humans to be unpleasant.  Although foul smelling, it occurs naturally and Exeter University's Dr Mark Wood has said: “Although hydrogen sulfide is well known as a pungent, foul-smelling gas in rotten eggs and flatulence, it is naturally produced in the body and could in fact be a healthcare hero with significant implications for future therapies for a variety of diseases.”  Studies have been carried out to determine the cause of smelly farts and to understand why they smell so repugnant. It has been theorized that certain foods cause a normal fart to become a smelly fart, particularly if the subjects digestive system is not accustomed to it. However there is variation between how different people react to the same food due to the unique makeup of each person's digestive system.   
In society and culture
Farting in public is considered to be embarrassing in most countries, as opposed to ordinary flatulence, smelly farts are particularly unpleasant to those in the vicinity, which has been shown to acutely increase the subjects embarrassment.  The social conventions pertaining to smelly farts varies by area, in certain situations and places, farting is not considered polite or normal, and a smelly fart is cause for great concern to the person emitting it.  However in other countries, farting is not considered to be a concern, but rather a slightly unpleasant smell.  A study by Dr Ajit Shukla, a consultant urologist at Metro Hospital, Delhi, found that it is normal to 'pass gas' when urinating or 'going to the toilet', in private this is a non-issue, however when using public facilities this can be a situation of embarrassment. 
Metro has rated smelly farts as the no. 1 worst thing to happen whilst copulating.  Many people do not know how smelly farts form, which is the main reason for unexpected situations of passing gas which has a unusually repugnant odor without anticipating it.