A single-longitudinal-mode (SLM) Brillouin erbium-doped fiber laser (BEDFL) with multiple-ring cavities (MRC) and fiber Bragg grating tunable filter (FBG-TF) is proposed and demonstrated. The fine structure of optical spectrum is measured by observing the laser beat signals in the experiment, and effects of Brillouin gain, FBG-TF, and MRC on the single-mode output are discussed, respectively. The output characteristic curve and wavelength stability test chart are given. The single-mode output power of 4 dBm with the signal-noise-rate over 60 dB and linewidth of less than 1.5 kHz is achieved.