Brown noise is a random noise which mimics the signal noise produced by brownian motion.
Hear a 5-second example of brown noise (download file).
Technically speaking, the spectral density is proportional to 1/f2, which basically means it has more energy at lower frequencies (decreasing by around 6dB per octave).
To the human ear, brown noise is similar to white noise but at a lower frequency. Examples in nature include waves on the beach and some wind noise.
Note: Some people use the term brown noise as a synonym for brown note, a controversial and unproven sound which causes the listener to lose control of their bowels.
Main page: Noise Colours