We present spectroscopy and Hα imaging of a faint nebula surrounding the X-ray bright, nearby neutron star RX J1856.5-3754 . The nebula shows no strong lines other than the hydrogen Balmer lines and has a cometary-like morphology, with the apex being approximately 1" ahead of the neutron star, and the tail extending up to at least 25" behind it. We find that the current observations can be satisfactorily accounted for by two different models. In the first, the nebula is similar to "Balmer-dominated" cometary nebulae seen around several radio pulsars, and is due to a bow shock in the ambient gas arising from the supersonic motion of a neutron star with a relativistic wind. In this case, the emission arises from shocked ambient gas; we find that the observations require an ambient neutral hydrogen number density n_(H^0) ≃ 0.8 cm^(-3) and a rotational energy loss Ė ≃ 6 × 10^(31) erg s^(-1). In the second model, the nebula is an ionisation nebula, but of a type not observed before (though expected to exist), in which the ionisation and heating are very rapid compared to recombination and cooling. Because of the hard ionising photons, the plasma is heated up to ~70 000 K and the emission is dominated by collisional excitation. The cometary morphology arises naturally as a consequence of the lack of emission from the plasma near and behind the neutron star (which is ionised completely) and of thermal expansion. We confirm this using a detailed hydrodynamical simulation. We find that to reproduce the observations for this case, the neutral hydrogen number density should be n_(H^0) ≃ 3 cm^(-3) and the extreme ultraviolet flux of the neutron star should be slightly in excess, by a factor ~1.7, over what is expected from a black-body fit to the optical and X-ray fluxes of the source. For this case, the rotational energy loss is less than 2 × 10^(32) erg s^(-1). Independent of the model, we find that RX J1856.5-3754 is not kept hot by accretion. If it is young and cooling, the lack of pulsations at X-ray wavelengths is puzzling. Using phenomenological arguments, we suggest that RX J1856.5-3754 may have a relatively weak, few 10^(11) G, magnetic field. If so, it would be ironic that the two brightest nearby neutron stars, RX J1856.5-3754 and RX J0720.4-3125 , may well represent the extreme ends of the neutron star magnetic field distribution, one a weak field neutron star and another a magnetar.