Different types of noise are generated by different stochastic processes. The power spectrum of a noise signal is referred to using colors.
One type is white noise, which is when each component of the noise signal have a probability distribution with zero mean and finite variance, and are statistically independent. This results in a noise signal with spectral density that is even throughout all frequencies (flat power spectral density). Note that the name is drawn from the white light as it contains all colors.
Another type is red noise or Brownian noise, which refers to noise resulting from Brownian motion. The spectral density of this type is inversely proportional to the frequency squared. Meaning, its power drastically decreases as its frequency increases (has more energy at low frequencies). Note that it is called red noise as it is analogous to red light which has a low frequency.
I found the figure below here. It might help better convey the difference in the power spectrums.
One more thing I forgot to mention that could clear your confusion. First off, please note that red noise and Brownian noise are not synonyms. All Brownian noise are red noise but not vice versa. Brownian motion has a Gaussian probability distribution. In other words, $B(t)$ is a Gaussian random variable with mean $0$ and variance $t$. In this sense, white noise can be thought of as the derivative of a Brownian motion. This has more details to it but I'm just trying to highlight the relation between white noise and Brownian motion.