The Copenhagen interpretation is named after the city that Niels Bohr and Werner Heisenberg worked on quantum mechanics together. The term refers to the set of mathematical principles that had been worked out by the pair during the 1920's. The term itself was not coined until later, when Heisenberg used it.
The relevant principles of this interpretation are as follows:
Disclaimer, as I am not a quantum physicist, the following explanation is about to get very vague.
What the this interpretation has to say about the double slit experiment is that a given particle (could be light or matter) exhibits the properties of both particles and waves. The particular bit that makes the experiment work, is that they cannot be both at the same time. They change their type based on the method used to measure them. In the version of the experiment that has a detector at the slits causes the photon or electron or whatever to behave like a particle because it was measured like a particle. In other versions of the experiment, where there exists no detector at the slits, the photon or electron or whatever is measured like a wave, so it behaves like a wave.[1]
Note that when I say "photon or electron or whatever" I am not being unintentionally vague. What I mean to say is that matter exhibits these properties as well. What this means is that while large objects seem to be well defined, in reality they are much less well defined. The classical idea of objects having a clear difference between them starts to break down, and it appears that everything is less defined then we thought.