Here's a short explanation of the experiment reported in "Quantum imaging with undetected photons" by members of Anton Zeilinger's group in Vienna (Barreta Lemos, Borish, Cole, Ramelow, Lapkiewicz and Zeilinger). The previous post also explains the experiment, but in a way that is closer to my real-time reading of the article; this post is cleaner and more succinct.
It's most easily understood by comparison to an ordinary Mach-Zehnder interferometry experiment. (The most informative part of the wikipedia article is the section "How it works"; Fig. 3 provides a picture.) In this sort of experiment, photons from a source such as a laser encounter a beamsplitter and go into a superposition of being transmitted and reflected. One beam goes through an object to be imaged, and acquires a phase factor---a complex number of modulus 1 that depends on the refractive index of the material out of which the object is made, and the thickness of the object at the point at which the beam goes through. You can think of this complex number as an arrow of length 1 lying in a two-dimensional plane; the arrow rotates as the photon passes through material, with the rate of rotation depending on the refractive index of the material. (If the thickness and/or refractive index varies on a scale smaller than the beamwidth, then the phase shift may vary over the beam cross-section, allowing the creation of an image of how the thickness of the object---or at least, the total phase imparted by the object, since the refractive index may be varying too---varies in the plane transverse to the beam. Otherwise, to create an image rather than just measure the total phase it imparts at a point, the beam may need to be scanned across the object.) The phase shift can be detected by recombining the beams at the second beamsplitter, and observing the intensity of light in each of the two output beams, since the relative probability of a photon coming out one way or the other depends on the relative phase of the the two input beams; this dependence is called "interference".
Now open the homepage of the Nature article and click on Figure 1 to enlarge it. This is a simplified schematic of the experiment done in Vienna. Just as in ordinary Mach-Zehnder interferometry, a beam of photons is split on a beamsplitter (labeled BS1 in the figure). One can think of each photon from the source going into a superposition of being reflected and transmitted at the first beamsplitter. The transmitted part is downconverted by passing through the nonlinear crystal NL1 into an entangled pair consisting of a yellow and a red photon; the red photon is siphoned off by a dichroic (color-dependent) beamsplitter, D1, and passed through the object O to be imaged, acquiring a phase dependent on the refractive index of the object and its thickness. The phase, as I understand things, is associated with the photon pair even though it is imparted by the passing only the red photon through the object. In order to observe the phase via interferometry, one needs to involve both the red and yellow photon, coherently. (If one could observe it as soon as it was imparted to the pair by just interacting with the yellow photon, one could send a signal from the interaction point to the yellow part of the beam instantaneously, violating relativity.) The red part of the beam is then recombined (at dichroic beamsplitter D2) with the reflected portion of the beam (which is still at the original wavelength), and that portion of the beam is passed through another nonlinear crystal, NL2. This downconverts the part of the beam that is at the original wavelength into a red-yellow pair, with the resulting red component aligned with --- and indistinguishable from---the red component that has gone through the object. The phase associated with the photon pair created in the transmitted part of the beam whose red member went through the object is now associated with the yellow photons in the transmitted beam, since the red photons in that beam have been rendered indistinguishable from the ones created in the reflected beam, and so retain no information about the relative phase. This means that the phase can be observed siphoning out the red photons (at dichroic beamsplitter D3), recombining just the yellow photons with a beamsplitter BS2, and observing the intensitities at the two outputs of this final beamsplitter, precisely as in the last stage of an ordinary Mach-Zehnder experiment. The potential advantage over ordinary Mach-Zehnder interferometry is that one can image the total phase imparted by the object at a wavelength different from the wavelength of the photons that are interfered and detected at the final stage, which could be an advantage for instance if good detectors are not available at the wavelength one wants to image the object at.