"Colour" is essentially a property of the distribution of wavelengths of visible light (as perceived by humans).

Digital cameras only detect the amount of light at each pixel, they can't measure the wavelength and thus can't record colours directly. Colour images are produced by placing alternating red/green/blue filters in front of each pixel. By placing a red filter (one that blocks green and blue light) in front of a pixel you can thus measure the amount of red light at that location.

Infra-red photography with standard digital cameras involves filtering out visible light (and optionally removing the built in IR filtering) so only infra-red light is recorded. The alternating red/green/blue filters remain in place.

There are different wavelengths of infra-red light, however these wavelengths don't correspond to "colour" because they are invisible to the human eye. True infrared, in the 850nm and longer range passes more or less equally through each of the red/green/blue filters so you end up with an intensity only (greyscale) image, like this:

http://www.mattgrum.com/photo_se/IR_1.jpg

Wavelengths that are closer to the visible spectrum, so call near IR in the 665nm range will pass through the RGB filters in different amounts so an image with different RGB values is produced and hence when displayed on computer you get a colour image.

---

But the colours aren't "real", in the sense that colour is an property of human vision and these wavelengths are outside our vision so the brain hasn't defined a way of presenting them to us. The different colours you see in a digital infrared image (reproduced in the visible range by your computer monitor) arise due to a deficiency in the blue and green filters.

The blue filters are designed to filter out the lower frequency red and green light, but around the visible spectrum range (as the camera's IR filter normally takes out everything else). When visible light is blocked and frequencies get really low (like those reflected by foliage via the Wood Effect) they start to pass through the blue and green filters again!

So the very bottom of the visible spectrum/very near IR (which is plentiful in the sky) mainly excites the red pixels as the blue and green filters are still doing their job, near IR (reflected from leaves) starts to excite blue and green pixels as the filters are operating outside their normal range.

The result is a red looking sky and blue/turquoise looking trees, like this:

(source: wearejuno.com)

But since these colours aren't reall real, photographers often swap the red/blue channels around, which gives more normal looking blue skies and green/yellow trees:

http://www.mattgrum.com/photo_se/IR_2.jpg