Why are two independent sources incoherent? Coherent sources are produced from a single parent source. But, why are two independent sources always incoherent? Two sources can produce light of the same frequency. Then, I guess the problem is with phase. Are two independent sources always out of phase? why so?
 A: We usually generate light by black body radiation, that is by heating something until it glows. There are many sources for black body radiation, but the dominant one is usually random thermal motion causing random transient electric dipoles within the black body. The changes in these dipoles generates eletromagnetic radiation, and because the dipole changes are random so is the EM radiation they generate.
If you take two points in the black body that are close compared to the wavelength of the lattice vibrations then their motion, and hence the EM generated, will be closely correlated. However as you increase the separation between the two points the correlation will decrease and at macroscopic distance will be essentially zero.
A: 
What is coherence?
In physics, coherence is an ideal property of waves that enables stationary (i.e. temporally and spatially constant) interference.   

So, sustained interference is necessary for coherence.  
In practice, a sustained interference pattern cannot be obtained by using two independent sources of light. It is because of the following reasons:  


*

*Two independent sources of light can't emit waves continuously.

*The waves emitted by two independent sources of light do not have same phase or a constant phase difference.  


Even how far would one expect stable interference form a single source, because light is emitted by each excited atom of the source, which are billions in number. Any way your question is not that, lets not go in deep with it. 
I will explain with the classic example given by my teacher. If you assume that an excited atom emits light in a time of the order of $10^{-8}$second. And if you two independent sources, then the waves from two points in two sources will have a definite phase relationship only for $10^{-8}$second and accordingly interference pattern formed by them will last on the screen only for this much time. The next bursts of waves coming from the two points in the two sources may be having an altogether different phase relationship and consequently the interference pattern may get shifted to some other place on the screen. These changes in positions in maxima and minima of intensity of light will occur about $10^{8}$ times in one second. As the changes are too fast to be followed by the human eye, we shall not observe pattern and there will be general illumination over the screen. So, there will be no stationary interference and thus two independent sources can't be coherent in practice. If it is possible, it is a ideal case.     
Response to the comment of Gatsu. 

Radiations are produced due to rapid acceleration and decelerations of electrons (radio wave), Klystron valve or magnetron valve (micro wave), vibration of atoms and molecules (infra-red wave), transition of electron from shells (light), etc. As your question is concerned only to visible light (only part of the spectrum that is detected by the human eye), I will consider radiation being emitted from the electrons in the atoms of Sun when they move from one energy level to a lower energy level.   
If you drop a stone into the water, sphereical waves (click here) will be formed. If you look at those waves by other side, you will get to see some what as shown above. Dark lines (wave fronts) what you are seeing in the above figure, are stretched into concentric circles to get the appearance of spherical wave fronts (click here).  
As like waves are formed when you drop a stone into the water, you can visualize a spherical wave being formed when an electron jumps from higher energy state to lower energy state. As we are only dealing with waves emanating in the forward direction, lets not worry about the wave emanated backwards. Atom is so small (about the order of $10^{-10}$ in radius), so you can imagine how small the radius of electron will be (particle nature of electron has been considered), thus you can also can visualize how small spherical primary wave front radius (distance of from any point on the wave front to the center of electron) will be. According to Huygen's priciple each point on the given or primary wave front acts as a source of secondary wavelets, sending out disturbance in all directions in a similar manner as the original source of light. So, by the time the spherical wave front reaches your slits of original double slit experiment, the radius of that reached spherical wave front will be big enough that the same wave front enters through both the slits, so you will see the interference pattern even from the sun light. 
