What do the colors in false color images represent?

Part of why you don't see colors in astronomical objects through a telescope is that your eye isn't sensitive to colors when what you are looking at is faint. Your eyes have two types of photoreceptors: rods and cones. Cones detect color, but rods are more sensitive. So, when seeing something faint, you mostly use your rods, and you don't get much color. Try looking at a color photograph in a dimly lit room.

As Geoff Gaherty points out, if the objects were much brighter, you would indeed see them in color.

However, they still wouldn't necessarily be the same colors you see in the images, because most images are indeed false color. What the false color means really depends on the data in question. What wavelengths an image represents depends on what filter was being used (if any) when the image was taken, and the sensitivity of the detector (eg CCD) being used. So, different images of the same object may look very different. For example, compare this image of the Lagoon Nebula (M8) to this one.

Few astronomers use filter sets designed to match the human eye. It is more common for filter sets to be selected based on scientific considerations. General purpose sets of filters in common use do not match the human eye: compare the transmission curves for the Johnson-Cousins UBVRI filters and the SDSS filters the the sensativity of human cone cells. So, a set of images of an object from a given astronomical telescope may have images at several wavelengths, but these will probably not be exactly those that correspond to red, green, and blue to the human eye. Still, the easiest way for humans to visualise this data is to map these images to the red, green, and blue channels in an image, basically pretending that they are.

In addition to simply mapping images through different filters to the RGB channels of an image, more complex approaches are sometimes used. See, for example, this paper (2004PASP..116..133L).

So, ultimately, what the colors you see in a false color image actually mean depends both of what data happened to be used to be make the image and the method of doing the mapping preferred by whoever constructed the image.

Answering that the colours are false is wrong. False colours are used only in a small minority of astronomical photographs. In most cases, the colours are 100% real. They certainly haven't been added by computers, as some people claim. The first colour photographs of astronomical objects came out in the late 1950s, and showed brilliant reds and blues. This was decades before computers began to be used in astrophotography.

The correct answer is that the colours are real, but the human eye lacks the capability of seeing any colours at such low levels of light intensity. The colours are there, but everything is interpreted as shades of grey green by the human eye.

I can count only three times in 54 years of observing when I have seen colours in deep sky objects, and all were with very large apertures: 18-inch (Eta Carinae Nebula), 22-inch (Dumbbell Nebula), and 74-inch (Cat's Eye Nebula).

For solar physics, the false colors were used to quickly identify the filter that was used, and possibly even the instrument itself.

So, for instance, SOHO/EIT, there are four filters, each one typically shown with a color that are ordered by spectrum (eg, the 'green' false color image has a spectral sensitivity between the 'yellow' and 'blue' images. 'yellow' is between 'orange' and 'green')

STEREO/SECCHI/EUVI used the same colors for the corresponding spectral lines that their filters were sensitive to, so when you'd see a 'blue' picture of the sun, you knew it was near 171 Angstrom, 'green' was near 195 Angstrom, etc.

And then came SDO/AIA, which was done by the team who brought you TRACE, so they carried over the colorings from their previous instrument, which never generated full disk images (unless as a mosaic). So now, the 171 images are yellow, not blue. The Blue images are actually 335, which would be closer to the red/orange 304 images, which themselves have enough disagreement on color tables that even the SDO mission website uses a table closer to the EIT/EUVI tables than the AIA PI teams' table. (so, in a way, the color table also reflects what the scientist is more interested in ... Flaring regions, or the everyday stuff.)

update : I should also mention that the preferred term when discussing color mappings on single-wavelength images is 'coded color', especially for spectral ranges outside the range of human vision. 'False color' is still appropriate when you mix three wavelengths together as RGB channels.