This version of the experiment comes out of the previous version, where particles are released one by one into a detector. This time, the question is asked, what is the "motion" (I understand that this is a somewhat incorrect term to apply to sub-atomic particles, but if Feynman can do it, so can I [1]) of the particle through the two slits. Is the particle passing through both slits?
After the confusing results of the previous dual slit experiment, physicists decided to create a new version of the dual slit experiment, this time with an interferometer positioned at the slits to help determine which slit the particle goes through. As Feynman puts it in the video linked above, "Lets squeeze nature into some kind of a difficulty".
The experiment is performed as before, except there is the addition of a device to detect a particle passing through one slit or the other (or both). For the purposes of this discussion, the details of the device don't matter, since the results are the same no matter the method of measurement. At the time, there were questions of if the method mattered, but we since know that any method to determine the position will produce the same results.
Again, particles are released slow enough to be one at a time, and the probability of a particle hitting a given spot is measured (via the incidence rate like before). The hope is that this experiment will allow us to see more into what is happening with particles to produce this interference pattern.
The results of the experiment are frustrating. What happens when measure the particles position we get the pattern we would expect if the particle was indeed a particle. Essentially, the particle is no longer behaving like a wave.
The results of this experiment have led to many new ideas, since the classical way of thinking about these problems simply doesn't work[2]. To explain this phenomena, there exist many methods. These methods include some strange ideas in order to explain this, among which is the many worlds interpretation, which basically says that there are other worlds that cross over at the quantum level. I would like to note that I'm not endorsing this theory in anyway, just using it to point out the difficulty that humans have with grasping the concept.
The more popular interpretation is the Cophenhagen interpretation, which is discussed on its own page