Nonlinear vibration absorbers, commonly referred to as Nonlinear Energy Sinks (NESs), have been the object of several theoretical and experimental studies over the past decade. This work illustrates the theoretical design and experimental realization of a Nonlinear Energy Sink coupled to an energy harvester. The mass of the Magnetic-Strung NES is a magnet which is linked to the primary system by means of two strings with adjustable pretension that work transversally. The restoring elastic force of the strings is modulated by the magnetic force applied by two magnets suitably located on the primary mass. Either a cubic or a bistable configuration may be obtained, depending on the distance of the additional magnets, NES's efficiency as an absorber is studied on a harmonically forced single degree-of-freedom primary system. The Target Energy Transfer (TET) from the primary system to the NES, as well as different response regimes like the Strongly Modulated Response (SMR), are experimentally observed. Furthermore, the harvesting of energy from the NES vibrations is also investigated by coupling the mechanical system with a coil for electromagnetic energy conversion. Consequently, the vibration energy of the primary mass is absorbed by the NES and finally converted into electric energy.