If photons are fired through the experiment one at a time (firing photons at a wall with two holes and a photon detector on the other side of the holes), they will build up an interference patter on the other side, as if they had gone through both holes at once and interfered with themselves. If the experiment is set up so that detectors monitor which hole the photo goes through, the photon is indeed observed to be going through only one hole, and there is no interference pattern. If a detector is set up not at the holes but intermediate between the two holes and the back wall detector screen then it may be possible to see which route a particular photon took after it had passed the two holes before it arrived at the screen. Quantum theory says that if we choose to turn this new detector off and not look at the photons, they will form an interference pattern. But if we look at the photons to see which hole they went through, even if we look after they have gone through the hole, there will be no interference pattern. The delayed choice comes into the story because we can make the decision whether or not too look at the photon after the photon has already passed through the hole[s]. The decision made seems to determine how the photon behaved at the time it was passing though the hole a tiny fraction of a second in the past. It seems as though the photons have some precognition about how the set-up of the experiment will be before it sets out on its journey. This has also provided credence to the metaphysical concept of backwards causation.
See John R. Gribbin, Mary Gribbin, and Jonathan Gribbin (Q Is for Quantum: Particle Physics from A-Z. London: Weidenfeld & Nicolson, 1998), 102-103.