In talk about intelligence, the concepts in the title above 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 who, by many people and over a long period of time, is considered to have contributed a great deal in a particular field and to have been of lasting influence.
So, a genius is not just someone who is very good in his field, it is not someone above a certain I.Q. (no matter how high), it is not a savant, it is not a prodigy, and it is not a gifted. 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 to have an I.Q. score at or above the 98th centile of the adult population. As tests do not correlate perfectly, this means that a bit more than 3 % of the population are 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 achieve that score 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 the playing, drawing, mathematics et cetera). 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; in the case of child prodigies, they may be precocious rather than gifted). 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 also 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, however, am 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 one is doing. Savants are mostly retarded, hence the old term idiot savant, and often have autism in combination with retardation. Someone who speaks about oneself as being a savant is likely not a savant. It is a mistake to believe that autism always goes with being a savant; only part 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.