In talk about intelligence, the four concepts in this article's title are eternally confused with each other. So once and for all:
This concept was defined thoroughly by Francis Galton. His definition boils down to (in my own words): A genius is someone of extraordinary ability who, by many people and over a long period of time, is considered to have contributed a great deal in any field and to have been or be of great influence.
As I personally find that definition unsatisfactory (it is saying the same twice) I am developing one of my own (see the relevant writings by me).
In any case, a genius is not someone who is very good (no matter how good) in his field, it is not someone with an I.Q. above 140 (or whatever number), it is not a savant (see below what that is), it is not a prodigy (see there) and it is not a gifted (see there). Obviously a genius may by coincidence be any of those (except a savant) next to being a genius, but those words do not have the meaning of "genius".
Being gifted, following the most widely accepted definition, means you have an I.Q. score at or above the 98th centile of the adult population. And that is all. As tests do not correlate perfectly, this means a bit more than 3% of the population is gifted.
Whether children who score at or above the 98th centile against age peers are to be considered gifted as well is doubtful (I personally say no); but when they score that against adult norms they are certainly gifted. So children can be identified as gifted, but I recommend against relying on age peer norms for that.
A gifted is not just someone who is very good in some field (violin playing, drawing, math etcetera). Yes, such people are often called gifted, but that is a narrower sense of the word than is meant here. A gifted is not a genius, prodigy or savant, although a gifted may by coincidence be any of those (except a savant) as well. See also the article Reasons to avoid the term "gifted".
A prodigy is a child (or adult, if the word is used in a broader sense) who is exceptionally good at something. Prodigies are usually gifted, but not always; they may be precocious rather than gifted (see the explanation for "gifted" above). In any case they are not retarded, for then they would be called savant and not prodigy. Not all prodigies grow up to be exceptionally good in their childhood field of excellence.
A prodigy is not a genius (but may in rare instances grow up to be one) and not a savant. Whether one is recognized as a prodigy partly depends on how perceptive one's childhood environment is.
When a prodigy does not grow up to be a genius this is often said to be a tragedy; I personally though am more inclined to say the prodigy was precocious rather than brilliant in such a case.
A savant is someone who is or appears to be exceptionally good at one or a few narrow skills, without having the general intelligence to put that skill to practical use or even understand what he or she is doing. Savants are mostly retarded, and often have autism in combination with retardation. It is a mistake though to believe autism often goes with being a savant; only a small percentage of autistics are savants. A cow is an animal, but not all animals are cows. Autistics do have "circumscribed interests", but that is no reason to qualify them as savants.
A savant is never a genius, gifted or prodigy.