Different colors of noise are named by analogy with colors of light. Pink noise is between white noise and red noise.
White noise has equal power at all frequencies, just as white light is a combination of all the frequencies of the visible spectrum. The components of red noise are weighted toward low frequencies, just as red light is at the low end of the visible spectrum. Pink noise is weighted toward low frequencies too, but not as strongly as red. Specifically, the power in red noise drops off like 1/f² where f is frequency. The power in pink noise drops off like 1/f.
Generating pink noise is more complicated than you might think. The book Creating Noise, by Stefan Hollos and J. Richard Hollos, has a good explanation and C source code for generating pink noise and variations such as 1/f α noise for 0 < α < 1. If you want even more background, check out Recursive Digital Filters by the same authors.
If you’d like to hear what pink noise sounds like, here’s a sample that was created using the software in the book with a 6th order filter.