Color-temperature is a way to describe the color of light along a spectrum that goes from warm colors (measured as having a lower temperature) to cool colors (measures as having a higher temperature).

Color-temperature is measured in kelvins and corresponds to the temperature at which a certain metal must be heated to emit light of that color. That is why lower temperatures (say 3000 K) give off warm (yellow-orange) light and that high temperatures (9000 K) give off color (blueish) light.

White-balance is the process of canceling the effect color-temperature in a photograph. This is because our brain corrects what our eyes see to let us see white as white, although it may be tinted by the color-temperature of light.

If the color-temperature of light is known, then its effect can be mostly canceled. Automatic white-balance guesses at that temperature (see this question). Preset white-balance use mostly fixed known temperatures (6500 K for daylight, 9500 K for cloudy, 3000 K for tungsten, etc - actual values vary between cameras).

For custom white-balance the camera uses a sample that is known to be white and deduces the color-temperature of light from that. Some cameras actually report the measured color-temperatures when using custom white-balance.

Note that for light sources that are mixed or simply way off from typical lighting conditions (sodium vapor lights, some fluorescents) it is not possible to cancel the effect because some parts of the visible spectrum are missing.

Since colors affect the mood of a photograph and how we perceive images, so does color-temperature. It is known that warmer colors (lower temperatures) are generally perceived as more pleasant and some photographers purposefully set the white-balance wrong to convey a certain mood. Cool colors (higher temperatures) are associated with night and mystery. Neither is better, it must simply fit the photo and how you intend viewers to perceive it.