White noise vs pink noise vs brown noise
They're all steady, random sounds. The difference is how much of the sound sits in the high notes versus the low ones, and that changes how they feel to sleep with.
White noise
White noise has equal energy at every frequency. Because our ears are more sensitive to high pitches, it comes across as bright and hissy, like a detuned radio or TV static.
It's very good at masking other sounds, which is why it's used in offices and baby monitors, but some people find it harsh for a whole night.
Pink noise
Pink noise turns the high end down, so each octave gets the same amount of energy. It sounds softer and more balanced, like steady rain or wind in the trees.
Brown noise
Brown noise (also called red noise) turns the highs down even further. What's left is a deep, low rumble, like a strong waterfall, heavy surf or the inside of a plane.
Many people who find white noise too sharp love brown noise for sleep and focus.
Where does a fan fit?
A fan isn't a lab-made noise color, but its sound sits on the deep, soft side: closer to brown or pink than to white. That's a big part of why a fan is so easy to sleep with. It masks noise well without the hiss.
Nature sounds land in the same range. Rain is roughly pink, while surf and strong wind lean brown.
Which should you choose?
Try them. If you want maximum masking and don't mind a bright sound, white noise works. If you want something gentler, pink noise or rain. If you like a deep hum, brown noise or a fan.
You can also layer: a fan underneath for steady coverage, with rain or waves on top for texture.