I'm not sure how much quantum mechanics you've done, I'm guessing you haven't yet learnt about unitary state evolution through Schrödinger's equation and measurement through quantum observables; please correct me if I'm wrong and I'll change my answer accordingly.
First of all, assume you are dealing with a classical wave and then read through the Physics SE answer How can a single slit diffraction produce an interference pattern? and the two articles Section "Single-slit Diffraction" on the Wikipedia page for Diffraction and the Hyperphysics "Fraunhofer Single Slit" Page linked in the comments. These articles will let you understand how something that fulfils the D'Alembert wave equation can show interference when it diffracts from a single slit.
Now for the quantum/wave/particle bit. For now forget about waves, particles and photons and I even want you to forget about the concept of "space" for a minute and and understand that there is only one object that begets and shows all the optical behaviours we witness: the second-quantised electromagnetic field.
The only things that are believed to be real in modern physics are this field and other quantum fields like it. There are only a handful of them. When we witness physical phenomena we are seeing interactions between these quantum fields.
The second quantised electromagnetic field can be thought of as a infinite gathering of quantum simple harmonic oscillators, one for each classical plane wave mode of Maxwell's equations. The eigenstates of quantum simple harmonic oscillators are discrete and they are evenly spaced by an amount of energy $h\,\nu$, where $\nu$ is the frequency of the oscillator in question. So each oscillator can change its state disk continuously, by taking up or shedding a whole number multiple of this basic energy "chunk" $h\,\nu$. So the interactions of the electromagnetic field with the other quantum fields in the world is by way of these discrete packets. I like to think of these packets not so much as billiard balls but more like discrete data packets that are swapped between networks on the Internet, thus giving being to "stuff that happens" on the Internet. The quantum fields of the World talk to each other in discrete, chunky, communications, thus giving being to everything that we see happenning around us.
Where are these quantum oscillators? Remember we haven't even talked about space, I ask you to forget about it! The answer is that they are nowhere in particular and everywhere all at once! For the quantum fields I spoke of are the space around us. We don't need to deal with the mysterious concept of a "void" any more in physics (an idea that actually used to give me nightmares as a child): empty space is nothing more than what we see when the quantum oscillators of the quantum fields of the World are all in their ground states!
Now let's come right back to Earth and think about our single slit experiment when we do it with "one photon at a time in the experimental kit". A one photon state is now simply a quantum superposition of one photon states in the quantum oscillators that make up space around our kit. One photon states propagate exactly following Maxwell's equations: you can try your luck reading my answers here (How can we interpret polarization and frequency when we are dealing with one single photon?) and here (Electromagnetic radiation and quanta), to get more info on exactly how the one photon state fulfils Maxwell's equations, but it certainly does so just the same, and, to the best of our knowledge, it does so exactly. There is no approximation. To my mind, you can't get something "wavier" than something that fulfills Maxwell's equations: the vector components in free space all fulfill the dispersionless D'Alembert wave equation (or, equivalently, Helmholtz's equation, if we take the time-Fourier transform of our solutions and analyse them one frequency at a time). And yet we're talking about one "particle", one photon! So you can't have anything wavier than a photon and you can't have anything chunkier that a photon all at once. The quantum field idea really gets rid of the notion that light can't be a wave and a particle at once. It is both (and likely more, withal, but that is as yet undiscovered physics).
Actually, if you look at my other answers, one photon states behave, in a certain sense, exactly like classical light waves: there is a bijective map between the classes of classical solutions to Maxwell's equations and the class of one photon states, although the interpretation of the meaning of a Maxwell equation solution is a little different in both cases. One photon states are much more like classical states than few photon states: in the latter case we can see the weird behaviours of entanglement and other wholly quantum effects.
So, somewhat ironically, you are actually more likely to witness "classical behaviour" if you do make sure that only one photon is in the kit at once! One photon states diffract and interfere in their passing through slits exactly like the wave fields I asked you to read about at the beginning of my anser.
Here, though, is where one photon propagation differs from shining a strong laser through a slit. The probability amplitude to absorb photon at a given point is what is propagated by Maxwell's equations (actually it's a bit more involved than that, but that's the basic idea). So our one photon state diffracts through the system, and when it reaches the screen, the electromagnetic quantum field undergoes one of its fundamental interactions with the other quantum fields, in this case the Fermionic quantum fields that make up, say, the CCD array photographing the interference pattern. So you get a single point on the photographic screen: but if you keep sending one photon "click click click" at a time through the experimental kit, however slowly you do it, those little dots will slowly make up exactly the intensity pattern calculated from the classical Maxwell equations for the system.