-Tunable metasurfaces, whose functionality can be dynamically modified, open up the possibility of ultracompact photonic components with reconfigurable applications. Here we consider a graphene monolayer subject to a spatially periodic gate bias, which, thank to surface plasmons in the graphene, acts as a tunable and extremely compact metasurface for terahertz radiation. After characterizing its functionality, we show that it serves as the basic building block of an ultrathin complete absorber. In this subwavelength-thickness device, transmission and reflection channels are blocked and electromagnetic energy is completely absorbed by the metasurface building blocks. The proposed structure can be used as a modulator, and its frequency of operation can be changed by scaling its size or adjusting the doping level.