Optical Transients from gamma-ray burst sources, in addition to offering a distance determination, convey important information about the physics of the emission mechanism, and perhaps also about the underlying energy source. As the gamma-ray phenomenon is extremely diverse, with timescales spanning several orders of magnitude, some diversity in optical counterpart signatures appears plausible. We have studied the optical transient that accompanied the gamma-ray burst of 1997 May 8, GRB 970508. Observations conducted at the 2.5 m Nordic Optical Telescope (NOT) and the 2.2 m telescope at the German-Spanish Calar Alto observatory (CAHA) cover the time interval starting 3 hr 5 minutes to 96 days after the high-energy event. This brackets all other published observations, including radio. When analyzed in conjunction with optical data from other observatories, evidence emerges for a composite light curve. The first interval, from 3 to 8 hr after the event, was characterized by a constant or slowly declining brightness. At a later moment, the brightness started increasing rapidly, and reached a maximum approximately 40 hr after the GRB. From that moment, the GRB brightness decayed approximately as a power law of index -1.21. The last observation, after 96 days, mR = 24.28 +/- 0.10, is brighter than the extrapolated power law, and hints that a constant component, mR = 25.50 +/- 0.40, is present. The optical transient is unresolved (FWHM 0."83) at the faintest magnitude level. The brightness of the optical transient, its duration, and the general shape of the light curve set this source apart from the single other optical transient known, that of the 1997 February 28 event.