A numerical study of the flow-induced vibration of two elastically mounted cylinders in tandem and staggered arrangements at Reynolds number $Re=200$ is presented. The cylinder centres are set at a streamwise distance of 1.5 cylinder diameters, placing the rear cylinder in the near-wake region of the front cylinder for the tandem arrangement. The cross-stream or lateral offset is varied between 0 and 5 cylinder diameters. The two cylinders are identical, with the same elastic mounting, and constrained to oscillate only in the cross-flow direction. The variation of flow behaviours is examined for static cylinders and for elastic mountings of a range of spring stiffnesses, or reduced velocity. At least seven major modes of flow response are identified, delineated by whether the oscillation is effectively symmetric, and the strength of the influence of the flow through the gap between the two cylinders. Submodes of these are also identified based on whether or not the flow remains periodic. More subtle temporal behaviours, such as period doubling, quasi-periodicity and chaos, are also identified and mapped. Across all of these regimes, the amplitudes of vibration and the magnitude of the fluid forces are quantified. The modes identified span the parameter space between two important limiting cases: two static bodies at varying lateral offset; and two elastically mounted bodies in a tandem configuration at varying spring stiffnesses. Some similarity in the response of extremely stiff or static bodies and extremely slack bodies is shown. This is explained by the fact that the slack bodies are free to move to an equilibrium position and stop, effectively becoming a static system. However, the most complex behaviour appears between these limits, when the bodies are in reasonably close proximity, and the natural structural frequency is close to the vortex shedding frequency of a single cylinder. This appears to be driven by the interplay between a series of time scales, including the vortex formation time, the advection time across the gap between the cylinders and the oscillation period of both bodies. This points out an important difference between this multi-body system and the classic single-cylinder vortex-induced vibration: two bodies in close proximity will not oscillate in a synchronised, periodic manner when their natural structural frequencies are close to the nominal vortex shedding frequency of a single cylinder.