The temperature of the surface of the Sun (photosphere) is between 4500° - 6000° Kelvin. Inside the core, it's around 15.7 million degrees Kelvin.
In other types of stars (neutron stars, white dwarves, etc.), what is the temperature of these areas (though many do not have these same layers) and how to they compare to the temperatures of the sun?
This question has two parts:
Surface Temperatures
A very useful diagram which shows surface temperatures, and also gives you the temperature of any star you can observe is the Herzsprung-Russell Diagram, this one from le.ac.uk.
As you can see, the yellow of our own sun places it in the 4.5 kKelvin to 6 kKelvin, as noted in the question. This temperature is down towards the lower end of average. The main sequence, where most stars are, tops out about 20 kKelvin, and there are some up towards the 40 kKelvin region - they aren't shown here as they are much rarer.
White dwarves are a little hotter than our sun - between 6 kKelvin and 10 kKelvin.
Neutron stars are way off the main sequence - young ones can be over 1 MKelvin!
Core temperature:
Internally, the core temperatures are dependent on the mass of the star. In our sun, energy is delivered via the proton-proton chain mechanism, which occurs up to about 20 MKelvins, whereas more massive stars can use the Carbon-Nitrogen-Oxygen cycle - which happens from about 15 MKelvins upwards.
The differences are mainly down to convection and radiation differences - this extract from Wikipedia's Main Sequence page describes this in some detail:
Because there is a temperature difference between the core and the surface, or photosphere, energy is transported outward via radiation and convection. A radiation zone, where energy is transported by radiation, is stable against convection and there is very little mixing of the plasma. By contrast, in a convection zone the energy is transported by bulk movement of plasma, with hotter material rising and cooler material descending. Convection is a more efficient mode for carrying energy than radiation, but it will only occur under conditions that create a steep temperature gradient. In massive stars (above 10 solar masses) the rate of energy generation by the CNO cycle is very sensitive to temperature, so the fusion is highly concentrated at the core. Consequently, there is a high temperature gradient in the core region, which results in a convection zone for more efficient energy transport. This mixing of material around the core removes the helium ash from the hydrogen-burning region, allowing more of the hydrogen in the star to be consumed during the main-sequence lifetime. The outer regions of a massive star transport energy by radiation, with little or no convection. Intermediate mass stars such as Sirius may transport energy primarily by radiation, with a small core convection region. Medium-sized, low mass stars like the Sun have a core region that is stable against convection, with a convection zone near the surface that mixes the outer layers. This results in a steady buildup of a helium-rich core, surrounded by a hydrogen-rich outer region. By contrast, cool, very low-mass stars (below 0.4 solar masses) are convective throughout. Thus the helium produced at the core is distributed across the star, producing a relatively uniform atmosphere and a proportionately longer main sequence lifespan.
Here you read: "The temperature inside a newly formed neutron star is from around 1011 to 1012 kelvin."
According to McCook and Sion Spectroscopically Identified White Dwarfs catalog, the hottest White Dwarf is RE J150208+661224 with 170 kK.
I read somewhere that the coldest WDs have Teffs between 3000 and 4000 K. If the universe were old enough, the first WDs would be now Black Dwarfs as cold as the space around them, 3 K.
For non-degenerate stars, we have:
Possibly, the hottest known main sequence star is HD 93129 A with 52 kK. The hypothetical Population III stars could be hotter than that.
For comparison, Sun's temperature is 5778 K (wikipedia).
The coldest known main sequence star is possibly 2MASS J0523-1403 with only 2075 K. Dieterich's paper suggests that the coldest possible star could not be much colder than that, or else it would be not a star, but a Brown Dwarf.
For fusors (objects that fuse Hydrogen - stars - plus objects that fuse Deuterium - Brown Dwarfs), models predict that at the present age of the universe a BD would have cooled down to ~260 K (sorry for not remembering the reference now). Like WDs, BDs could be as cold as the space if the universe were old enough, I guess. Then, black dwarfs aside, it seems that it is secure to consider objects colder than 260 K as planets.
Note that all temperatures listed here excepting those of the Neutron stars are temperatures measured at the surface of these stars. Their centers are much hotter than that.
Finally, I forgot about another hypothetical objects like Quark stars, Q-stars, etc. I would not be surprised if (they really exist outside theory) that their central temperatures would be higher than 1012 kelvin.
What would be the temperature of a supermassive Black Hole?
