DISCRETE astronomical sources of photons in the TeV energy range are believed to be associated with regions in the relativistic outflow of particles and radiation from compact objects, such as neutron stars and black holes. The Bur from such sources, together with the timescales on which they vary, can provide strong constraints on the emission mechanisms. Here we report the observation of two dramatic outbursts of TeV photons from the active galaxy Markarian 421 (Mrk421). In the first outburst, which had a doubling time of about one hour, the flux increased above the relatively quiescent value by more than a factor of 50, briefly making Mrk421 the brightest TeV source in the sky. In the second outburst, which lasted approximately 30 minutes, the flux increased by a factor of 20-25. These data suggest that the emission region is extremely small-perhaps even smaller than our Solar System. This could prove challenging for current theoretical models of such emissions.