color shift is a general term for "unfaithful color rendition". I will touch here its photography meaning.

You have a color shift when the medium you are recording your image on (the film) does not match the color temperature of the light source that illuminates the object you are photographing. There, this is the abstract definition, abstractly correct and in desperate need of an example.

This whole node answers the old amateur photgrapher's questions: "Why did Mr. Grobnitz come out this green ?", "Where did my tan go ?" and "Why can't I get the damn flowers to look good ?".

An Example, to Clarify

First, you should know what color temperature is.
Let us take a piece of daylight film. We will use slide film, to make our life simpler for now.
This film, Kodak says, will render 5000 K light bouncing off a white piece of cardboard as, indeed, white: which means that a picture of said cardboard in the bright midday sunlight will be starkly white. But what does white mean, in this context ? It means that the cardboard reflects visible light without any preference, that's to say, if you were to draw a graph of its reflectance vs. the light frequency you would get more or less a flat line.
Suppose that you want to take a picture of a reddish t-shirt. What does red mean ? It means that the t-shirt reflects light with a certain preference for lower frequences in the spectrum. So what you get is reflected red light. You take a picture, and whoa ! it is indeed red.

Still with me ? No fainting yet ? Ok, now let us go inside in the kitchen, at night. We turn on our common incandescent bulb. Light whose color temperature is, oh, 3500 K pours forth.
We use this light to take a picture, on daylight film, of a white piece of cardboard. By this point it should not be a wonder to you: it comes out quite red. Voilà: color shift.

Does this mean that if we were to point a spectrograph at the red t-shirt in sunlight and the white piece of cardboard in lamplight we would see the same spectrum ? No.
You have to remember that the common human being is a trichromat. Film works the same way, by analyzing light into three components.
So, those two spectra are not the same, but film sees them as the same: red.

My eyes are not working properly !

Now you ask "why don't I see all these color shifts, then ?". Indeed, it does not matter if you are reading a book on the beach, by fluorescent lighting, under a halogen or in open shadow: the paper looks white.
Well, I have to tell you that it is all in your head: you know that paper is white (culturally), so your brain makes it look white. Actually, your color vision is heavily mediated by your preconceptions, and your brain does all it can to make things match.
But, film has no brain: it records (trichromatically) whatever falls on it. So, if you are in open shadow under 10000 K skylight (really blue !), that paper is cerulean.

How do I make white white ?

If you are shooting slides, you have to enter the madness that is color correction filters, CCs for friends: these colored filters (that you must choose according to light color) make things right.
For example, if you are shooting in open shadow, the light is too blue, which means that you would use a CC yellow filter (yellow and blue are complementary colors). How would you know that the light is too blue ? Well, experience helps, but for fine tuning you need a colorimeter.
For color negative film, you would correct that when the negative is printed: this is your second chance to make white white.
This means that when you print (we will not go into detail here, because color printing is fascinatingly messy) you do more or less the same that you were doing with CC filters; you take out the excess color.
Of course (a touch of realism here), this is all highly idealized: we are supposing perfect films, perfect filters and perfect papers. In reality, nothing is perfect - which means that when printing you get one color right and everything else falls where it may.
Printers in a hurry usually try to get skin tones right, and snow white - and let the rest be handled by the customer's brain. After all, do you really know what exact shade of mauve your coverlet is ? You don't. But you know the exact shade of your SO's skin (or at least you think you do).

I am all set ! Let's go take pictures !

I can just see you, with your little box of CC filters, your new colorimeter and your instructions to the printer. You think you have it down pat, all this color shit. Well you don't: now I will tell you why some times you simply lose.

Mixed lights

Mixed lights, unlike mixed drinks, can be a serious problem. Imagine a room, lit by a north facing window. Gorgeous winter light. Gorgeous Dutch girl, modeling for you.
You notice a dark corner, which would spoil your picture (whose title is "Gorgeous Caucasian Model in Gorgeous Winter Light in Empty White Room").
No prob: you set up a serious photoflood light (basically a bigass light bulb) to light up the dark corner.

And you lose. Why ? Light from the photoflood (3500 K) spills on the left side of the model's face, while the right side is lit by the skylight (anywhere between 8000 K and 10000 K). The same thing happens to her hands, her legs ... you get the picture. No single CC correction or filterpack when printing will make her the right color all over.
The solution with negative film could be differential filtering when printing, a difficult and time-expensive technique. If you are shooting slide film or maybe a whole movie, then you have to resort to tricks like putting a blue filter on the photoflood (to make it match the skylight), or putting a big yellow filter on the window itself, which looks bizarre but works well.

Bastard light sources

Some light sources, like fluorescent tubes have a discontinuous spectrum; others are nearly monochromatic, like sodium vapor lamps.
Fluorescents (especially cheap ones) usually produce a ghastly greenish dominant, that makes people look like corpses. That can be sort of corrected using a FL-D filter. But it is better to just avoid that kind of light.
Sodium vapor lamps are even worse: try looking at a blue sweater and a black sweater under that light some time. See any difference ?

Of course, the wily photographer uses these (and other) color shifts to his advantage, to render a particular mood or to create beauty.
Actually, night photography is all about color shifts: some cities may even look better and more interesting in a picture than they do in real life. All the multicolor jewel-like lights and facades are color shifts.

public thanks to the micro-picky Gritchka for suggesting useful corrections