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Vortex-induced vibrations have been studied experimentally with emphasis on damping and Reynolds number effects. Our system was an elastically-mounted rigid circular cylinder, free to oscillate only transverse to the flow direction, with very low inherent damping. We were able to prescribe the mass, damping, and elasticity of the system over a wide range of values, with the damping controlled by a custom-made variable magnetic eddy-current damping system. Special emphasis is put on a nontraditional parameter formulation. The advantages of this formulation are explained, and an important new parameter, effective stiffness, is introduced. Using this new formulation, the amplitude and frequency responses are only a function of damping, Reynolds number, and effective stiffness. We show the effects that damping and Reynolds number each have on the amplitude and frequency response profiles and make the interesting observation that changes in damping or Reynolds number have similar effects. The maximum amplitudes of our systems are studied in detail. We theoretically show that they should be functions of both damping and Reynolds number. This allows us to create constant-Reynolds-number curves of maximum amplitude over a large range of damping values, which we call a "generalized" Griffin plot. We also define maximum amplitudes in the case of zero damping as limiting amplitudes, and show that they are only a function of Reynolds number. We experimentally determine our limiting amplitude dependence on Reynolds number over the range 200 < Reynolds number < 5050. Discontinuities in the amplitude response profile are also investigated. The discontinuity between the initial branch and the large-amplitude, upper branch is studied in two ways. First, the time-averaged behavior is examined to understand what controls the discontinuity and look for damping and Reynolds number effects. Second, we track the cycle-by-cycle transient response through this discontinuous amplitude change, induced either by changes in the tunnel velocity or system damping. Finally, we also find a new discontinuity hysteresis region between the lower branch and the desynchronized region, which appears to be a low Reynolds number effect and is only seen in systems with Reynolds number < 1000.