Why are stars white?
It has more to do with physiology of the eye rather than the spectrum of light produced by stars.
Stars emit light over the full range of visible wavelengths. Hot stars emit more blue/violet light, cool stars emit more red light. The Sun is relatively neutral in that regard, so does not have a strong colouration, but many other stars in the sky have temperatures as low as 3,000K (should appear reddish) to 20,000K or more (should be blueish).
However, light received by the eye has to be reasonably bright to trigger the colour sensitive cells (the cones). So whereas the Sun appears to be yellow(ish) (or to be more exact your eyes perceive the mix of wavelengths as yellowish), there are relatively few stars that are bright enough to look distinctly coloured. Betelgeuse is an example that most people see as reddish. Arcturus is distinctly orange (to me).
But fainter stars all appear white because they are mainly being seen by the rod cells in your eye, which are not very colour sensitive (and have no sensitivity to light at the extremes of red and blue). This is known as scotopic vision.
APPENDIX: I have found a few interesting links that appear to confirm this line of argument. Firstly, one can predict what stellar spectra would look like to the eye under conditions where the cone cells are operational. Here we can see that many stars would appear either pink (K-stars) through orange (M-stars). G-stars like the Sun are just off-white, whilst hot stars are distinctly blue (see also here).
These calculations do not include the effects of the Earth's atmosphere which preferentially absorb and scatter blue light. However, the effect of this on the colours cannot be that serious (it will have no effect at all on rod cells since they are not colour sensitive). Cone cells are sensitive over the range 450-650 nm. The typical atmospheric transmission at zenith for these wavelengths ranges from 85% to 93% (see Fig.1 of Burke et al. 2010), but would reduce to perhaps 70% to 85% for an object only 30 degrees above the horizon. Even at this low altitude, the differential effect is quite small. The colours of stars would not be changed by the atmosphere until they got quite close to the horizon.
Stars would have different colours, ranging from pale blue through to an orangey-red, if they were bright enough to excite the eye's cone cells.
Short answer: Many of them are not.
Stars can be modeled as black-bodies. That means that the spectrum of light that they emit depends on their temperature, and the color of a star can be described as its Color Temperature. The color temperatures of stars can vary quite greatly, because stars have very different temperatures and emission spectras. One of the things that you'll notice about color temperatures and blackbody radiation is that "white" is actually a pretty imprecise term.
With that in mind, different stars are categorized by their different spectral types, where stars of different spectral types prominently emit visible light in different noticeable wavelengths.
Some stars, such asLa Superba, Betelgeuse, and Antares are known for their noticeably red colors, even with the naked eye.
Other stars, like Rigel, Fomalhaut, and Vega have a bluish-white color.
La Superba is known as the reddest star that we've ever found, but it's too dim to be seen with the naked eye. Every other star I mentioned above though, fall in the top 20 brightest stars. For that reason, more light from these stars reach your eye, and differences in color are more easily seen (for reasons well explained in Rob's answer). Even many stars which appear white in the naked eye usually have a non-white color when viewed in binoculars or a telescope. Albireo is probably the most brilliant example.
I would offer two concrete (although not very physical) reasons why most stars still appear white to you. First, you may not be looking very closely. I don't say this flippantly at all. Before I started observing the night sky, I was unaware of the patterns and structure that exists. I know people who don't even realize that the night sky changes throughout the night (and from month to month). I really just say this to suggest that if all you ever give the night sky a casual glance, you'll fail to see the full detail. But the more you observe, the more the patterns and detail begin to emerge.
Second, to build off Rob's answer, your eyes are actually pretty bad at figuring out color. To illustrate this point, think of the last time you were in a low lighting environment. You probably had a much easier time seeing shapes, and outlines of objects than you did seeing color of individual objects. For dimmer stars, only a minuscule amount of their light arrives at your eyes, and it's just not enough for your eyes and brain to make an accurate determination of color.
The sun is white. I've seen people say that it just looks yellow because the blue light is being scattered by the atmosphere. Since starlight is also scattered your question is still valid. But based on these pictures, the sun still seems to look pretty white through our atmosphere. My guess is that, since you can safely look directly at a star, you learned their color through first-hand experience. You learn the color of the sun primarily from being told what color it is, and maybe from looking at it during sunrise and sunset, when there's more atmosphere and it's redder.
The color of stars in general is controlled by blackbody radiation. As something heats up, it starts of reddish, then it gets yellowish, then it gets whitish, then it gets bluish, but it's all pretty washed out and anything between 3000 and 10000 kelvins is going to look pretty white.