Most false color images are made entirely for illustration, and the creation of them is more an art than a science.
If you want to make a false color image, you have to start with a series of black and white images. All the cameras attached to telescopes are effectively just photon counters - if a photon makes it through the telescope and into the camera, it adds 1 to the number of photons that arrive from that patch of the sky. Now, generally astronomers are more interested in specific colors of light than they are in the total amount of light of any kind that arrives at the telescope. In order to limit the kind of light that actually makes it to the camera, then, we put a filter in front of the camera. This works in the same way as red-blue 3D glasses and images; the red lens only lets through red light, and the blue lens only lets through the blue, so each eye gets a different picture, and your brain reconstructs the depth of the image.
Filters for astronomy purposes are usually of a very specific color that corresponds to a physical process that you’re interested in. So, if you’re interested in looking at where the ionized hydrogen is, you create a filter that lets in only the light produced by ionized hydrogen, stick it on your telescope. Then you can produce an image which counts all the photons from the hydrogen and their locations on the sky. This image, which is in black and white, is what an astronomer would use to determine where these elements are in the sky.
To make a false color image, you’d get a bunch of these black and white images that correspond to different elements, and order them according to the wavelength of the light the elements give off. You then give the bluest wavelength the bluest color, and the reddest wavelength the reddest color, and patch them all together. Some images are quite scientifically useful to see where elements appear together relative to each other, but most of the spectacular images we have of the universe come from more artistic manipulations.
The ones that you see on the Hubble Heritage site have been tweaked so that they are more visually appealing. Mostly this involves playing with the colors and saturation levels assigned to the original black and white pictures, but it can result in much more dramatic images than if you just plopped all the images on top of each other and colored them in.
Have a question of your own, or is something here unclear or making you curious? Feel free to ask!