The first thing you'd want to do is to take away the continuum of space, because humans understand better discreet space. Its easier to think of space in discrete chunks, that is I divide space in units of known conventional size. Let's take the example of LCD screens in computers: the LCD is made of many pixels put on a 2-dimensional matrix. You can tell if a line on the screen is longer than another an for all purposes here it behaves like normal space, respects the same math laws and so on.
But then, each pixel that makes the screen has 3 tiny devices that make it emit light in different spectrum ranges, that is RED, GREEN, BLUE (RGB for short). Each device for a given pixel and color takes one input in the range of 0..255 to know the intensity of the light it lets out, 0 stands for no light emitted and 255 for the brightest. One pixel can be bright red and have all the other columns shut - that is it's RGB are (RED: 255, GREEN: 0, BLUE: 0). It is unessential that a pixel with RED 255, GREEN 0, BLUE 255 is actually purple.
And now the essential: you could consider each of those color values as a new dimension. Your LCD screen has now 3 new dimensions, making up for a total of 5 dimensions. But now, another contraption: what if you could stack several LCD displays on top each other (and see through any depth you wish for)? You could then "see" 6 dimensions, right?
Now, if you could imagine that such an "LCD cube" takes part of a space with pixels infinite small, infinite range for all dimensions width, height, depth and color and there you have a continuum 6-dimensional space. Again not considering that two colors make another color (red and blue make purple as stated above), you could add more color coded dimensions. I don't think you could represent these in real life or in software, but hey, this is imagination! So, given an "N" number of colors, there you have it: N-dimensional space with color coded dimensions!
If you want to know more about colors and displays the best place to start is the browser using HTML and CSS. You can make a txt file with
<div style="background: rgb(255, 0, 255)"></div> as content and start experimenting with colors. Search the web for: [http://www.google.com/search?client=ubuntu&channel=fs&q=html+colors&ie=utf-8&oe=utf-8]HTML color and you'll find plenty.
Another example is thermal gradient which measures "How hot it is at a given distance from a hot object?". Measuring that in many points around a hot object makes a 3D heat map. In this case temperature gradient is a fourth dimension, representing it however still uses color codes (usually dark/green, red, yellow, white for cold, hot, hotter and respectively the hottest). Thermal gradient