I misunderstood your question at first. I thought you were asking about the detector downstream of the double slit, where the interference pattern is visible; every practical double-slit experiment includes such a detector. But instead you are asking about a hypothetical detector which could "tag" a particle as having gone through one slit or the other. Most interference experiments do not have such a detector.
The idea of "tagging" a particle as having gone through one slit or the other, and the realization that such tagging would destroy the double-slit interference pattern, was hashed out in a long series of debates between Bohr and Einstein. Most introductory quantum mechanics textbooks will have at least some summary of the history of these discussions, which include many possible "detectors" with varying degrees of fancifulness.
A practical way to tag photons as having gone through one slit or another is to cover both slits with polarizing films. If the light polarizations are parallel, it's not possible to use this technique to tell which slit a given photon came through, and the interference pattern survives. If the light polarizations are perpendicular, the it would be possible in principle to detect whether a given photon went through one slit or the other; in this case the interference pattern is also absent. If the polarizers are at some other angle, it's a good homework problem to predict the intensity of the interference pattern.