How can a single slit diffraction produce an interference pattern? How can a light passed though a single slit produce a similar interference pattern to the double-slit experiment? How does the diffracted wave produce the points of cancellation and reinforcement, if there is only one wave?
 A: One way of understanding this which has always had intuitive appeal to me is the so-called Huygens Principle which basically states that every point on a wavefront can be considered a point source for a new spherical wave, and that the evolution of the wavefront can be determined by superposing all of these spherical waves at later times.  The Wikipedia article that I linked to has some really nice pictures of this.
Diffraction effects can then be explained using this principle.  Imagine, for example, that you shine light through an extremely small slit, say a slit about the size of the wavelength itself, then when plane waves pass through this slit, the part of the wave that goes through the slit acts as a point source and generates a spherical wave, so the light diffracts.  
If the slit is larger, however, then the part of the wavefronts that pass through the slit act as multiple little points sources for spherical waves, and these spherical waves interfere with each other to give an interference pattern.  In this way, the diffraction pattern is very much like multiple slit interference, except instead of multiple slits, the wave front itself splits into a bunch of adjacent point sources that interfere with each other.
A: Why not? In terms of wave character, this can be explained easily using Huygens principle (as by Josh). Here's an image for on how the wavefronts emerge from both large and small slits...
                  
In case of the large slit, the wavefronts are more and hence they arrive as similarly plane waves. Still, the curve can be noticed at the edge. Thus, the spherical wavefronts are significant only when the slit width is comparable to the wavelength of the wave - leading to "diffraction".
Here's an image from Wiki on how the spherical wavefronts interfere with each other...


As quantum mechanics has a major role, this can also be explained via the particle-character (photons) of light using the uncertainty principle. When the width of the slit is slowly reduced, the pattern becomes wider and wider until it disappears completely.
Because, once we're nearing to knowing the position accurately, the uncertainty in momentum increases or vice versa. Anyways, you can't be sure how the photon hits the screen and where...
A: One researcher observed correctly that in a single slit experiment with monochromatic light, the 2 edges of the slit were very bright as if they were the sources of light also. That explains the interference patterns and the deviation from the straight line that we observed. The electrons at the edge of the slit are excited by the incoming light and they become sources of light also. There are now 3 sources of light: the original one coming through the slit and the light emitted by each side of the slit. The density of the light is constant at some point behind the slit and the screen is not indicating any light because of this constant density, not because light was cancelled.
A: 1) The ends of single slit act as two sources of light waves
2) the situation is similar to double slit interference experiment
3) thus in single slit also, interference pattern is seen
A: Although @joshphysics has produced a good answer it misses the point that the OP has only a vague idea as to what diffraction and interference are and whether there is a difference between them.
Indeed, Robert Feynman said, No one has ever been able to define the difference between interference and diffraction satisfactorily. It is just a question of usage, and there is no specific, important physical difference between them.
There are many example in Physics Stack Exchange of a discussion about this.
The OP has been careful in stating the question How can a light passed though a single slit produce a similar interference pattern to the double-slit experiment? with the use of the word similar.
Both arrangements produce a fringe pattern but a key difference is that for a single clit the central maximum is twice the width of the other fringes whereas for a double slit the fringes are of equal width and their intensity is modulated by a single slit fringe pattern.
Historically there was a need to differentiate the fringe pattern produced by a discrete number of sources (interference) and the fringe pattern produced by very many sources (diffraction).
For a single slit the analysis is done by assuming and very large number of very small sources across a slit via an integration whereas the simple analysis for a double slit is done via the summation of two terms.
A: If you look close enough you can notice interference pattern even in 1 slit water tank experiment
https://www.youtube.com/watch?v=BH0NfVUTWG4
according to Huygens Principle light should always interfere with itself and create interference patterns no matter how and where you shine it. The key here is the sharp edges of the slit, and what actually happens at those edges. It has to do with turbulent flow and vorticity. (talking about the water here) 
