S.H.I.E.L.D is a group of members involved in saving Earth. All of those characters are fictional characters from Marvel comics. Mutants from the X-Men books are also fictional characters from Marvel comics. So why aren't any mutants from the X-Men series also members of S.H.I.E.L.D?
In the comics (Earth 616, Ultimates, etc)
For the most part, the X-Men are considered a "rogue" group. They operate outside the law and generally eschew outside authority (X-Factor being the exception to this).
As such, most mutants - and especially X-Men, would have little interest in joining up with what is effectively a government controlled military group.
More recently, Alison Blaire, a.k.a Dazzler has been recruited to S.H.I.E.L.D as a former X-Man who may help provide insight into Scott Summers' line of thought.
However, she has been secretly replaced by Mystique.
In the cinematic/TV universe (Earth 199999)
The primary reason behind the separation is, as others have pointed out, an issue of rights. Rights to all mutant franchises and references to them, as well as the Fantastic Four and are owned by Fox, while Marvel holds rights to S.H.I.E.L.D and the Avengers and related properties (and Sony still owns rights to the Spider-Man). As such, the mutants do not currently exist in the Marvel Cinematic Universe - with a small pseudo-exception of Quicksilver and Scarlet Witch, who will not be mutants in the MCU.
Beyond that, a distinction must be made in regards to being a member of S.H.I.E.L.D, and a member of the Avengers. Currently, in the MCU, no super-powered being exists as a member of S.H.I.E.L.D - only as a member of the Avengers (remember that Black Widow and Hawkeye are merely highly trained agents with exceptional abilities).
Edit: The recent Captain America: The Winter Soldier trailer mentions Cap saying that he "joined S.H.I.E.L.D" at some point. Exactly when this happened is unclear. He worked with/for them during The Avengers, but I never got the impression that any of the Avengers aside from Black Widow and Hawkeye were official agents.
During the course of The Avengers movie, the council which governs S.H.I.E.L.D shows disapproval of counting on super-powered beings. While Director Fury and his men on the ground may be OK fighting alongside the Avengers, it is entirely possible that S.H.I.E.L.D's governing body may restrict super-powered beings from being actual agents of S.H.I.E.L.D.
Short answer: Lawyers.
Long answer: The Marvel company sold the film rights to all X-Men and (I think) all mutant characters to the Fox company when Marvel was going bankrupt in the 1990s.
Because SHIELD is connected to the Marvel Cinematic Universe, they don't want to add in characters that can't be connected to that universe, like Mutants or Spider-Man characters.
If you're referring to the Marvel Cinematic Universe, it's because of which film companies own the rights to different groups of characters: the rights to use the X-Men (names, characters etc) on film are currently owned by 20th Century Fox (who also have the rights to use the Fantastic Four), which means those characters can't be used in other companies films. Similarly, the movie rights for Spiderman are currently owned by Sony.
Others have correctly answered this, out of universe. Within the Marvel Cinematic Universe, however, there is one potential reason which hasn't been touched upon.
In the Ultimate Universe (which the Marvel CU is largely based upon) the Ultimates (the Ultimate version of the Avengers) was formed by SHIELD as a public-facing super-team. They tried hard to make the group popular and to keep them humanized. Sure, they had a legendary super soldier and Iron Man on staff, but even they were essentially normal people who had gained super powers through science (or so they claimed). Part of the research being done were studies into popularity and image - the Ultimates were a public relations effort as much as an anti-terrorist/superhuman group. Nick Fury even pointed this out when he indicated why there were no mutants in the initial line up.
Fury later turned out to be lying (or mistaken) as the Wasp is a mutant in the Ultimate Universe, but this fact was never publicly known.
Thus, it's possible that within the Marvel Cinematic Universe the same decisions were made: Mutants, as generally unpopular segments of the population, were not considered for the government-sponsored super group.
The real Fury used Wolverine as a in a what if issue on the helicarrier with Black Widow and they are friends. Wolverine was a spy, Shield is providing some protection to x-men. I would like to see a few to show up, in a few other issues mutants have shown up, maybe bring in Punisher to help out, War Zone Punisher.