In this paper we present a novel dual-band metamaterial absorber (MA), which is composed of a periodically arranged 2nd order Koch curve array and a metal ground separated by a dielectric spacer. By employing the fractal characteristic of space-filling, more compact unit cell with a size reduction of 17.5% has been achieved as compared with the conventional square-shaped MA. The dual-band operation is not originated from the hybrid or stacked methods as reported before, but from the two distinct resonance modes of the 2nd order Koch curves induced by the incident electromagnetic wave, and can be realized within a single unit cell. Due to its rotationally symmetric pattern, the absorptivity of the above presented MA is insensitive to the polarization of the incident waves and can perform well in a wide range of incident angles. The effective medium theory has been employed to investigate the underlying physical mechanism of the fractal MA, and good agreements between simulation and experimental results have been achieved.