Basic Terms
Before we launch into a discussion of color and contrast, let’s make sure we are using the same language. Three terms may be used in describing any given color.
First is Hue, how colors distinguish themselves from each other in the rainbow or in the spectrum. Yellow is different from orange and from red, etc. It is also how colors are differentiated around a color wheel.
Second is Saturation, the relative richness or dullness of a color. All colors in the spectrum are fully saturated (except at the ends, where they lapse into black). Dusty rose is less saturated (or more neutral) than pink. Olive is a less saturated version of yellow-green.
The third term is Value, which is the inherent lightness or darkness of a color. Going around a properly constructed color wheel, the lightest color is yellow and the darkest is blue.
Of the more than two million colors that a normal human eye can discern, each one can be described using these three terms, the way we could use three axis points to determine the position of something in three-dimensional space or the size of something using height, width and depth.
Legibility
When we can see and read what is written or perceive the elements of a picture, we call that legibility. Legibility is a function of contrast.
Of the three qualities of color we just defined—hue, saturation and value—the one that matters most for contrast is value. Saturation and hue hardly make a difference, especially when distance or small size is involved.
To sum up so far: legibility is a function of contrast, and contrast is a function of value.
In graphic design, nothing compensates for a lack of legibility. No concept. No style or fashion. No coolness factor. I repeat: Nothing compensates for a lack of legibility. Without it, no one even sees or correctly perceives your work. It is fundamental.
A good tool for accurately describing values is tint percentage, developed in the printing industry. Halftone tints cover an area with a small uniform grid of dots on a paper, often smaller than can be seen with the naked eye. In a 70% halftone tint, 70% of the paper area is covered in ink and only 30% of the paper has no ink. These halftone tint percentages should be second nature to all graphic designers. 
Contrast Differential
The amount of contrast between type and its background can be calculated by subtracting the lesser value percentage from the greater. This is the Contrast Differential. A good rule of thumb for excellent contrast is 60% difference or more. Black type on white paper is a 100% difference, the greatest contrast possible, whereas 35% contrast differential is minimal.
Is it physically possible to read below that threshold?
Yes, of course. But is it convenient or comfortable? No. And what is the function of graphic design? To aid communication, not impede it. Why would any sane designer try to make the printed message anything less than convenient or comfortable to read?
There is no good reason.
Viewing this from a distance of even a few feet shows the need for excellent contrast.
Adapted from Logo Design Theory: How Branding Design Really Works