The blazar Mrk 501 was observed at energies above 0.10 TeV with the MAGIC Telescope from 2005 May through July. The high sensitivity of the instrument enabled the determination of the flux and spectrum of the source on a night-by-night basis. Throughout our observational campaign, the flux from Mrk 501 was found to vary by an order of magnitude. Intranight flux variability with flux-doubling times down to 2 minutes was observed during the two most active nights, namely, June 30 and July 9. These are the fastest flux variations ever observed in Mrk 501. The ~20 minute long flare of July 9 showed an indication of a 4+/-1 minute time delay between the peaks of F(<0.25 TeV) and F(>1.2 TeV), which may indicate a progressive acceleration of electrons in the emitting plasma blob. The flux variability was quantified for several energy ranges and found to increase with the energy of the γ-ray photons. The spectra hardened significantly with increasing flux, and during the two most active nights, a spectral peak was clearly detected at 0.43+/-0.06 and 0.25+/-0.07 TeV, respectively, for June 30 and July 9. There is no evidence of such a spectral feature for the other nights at energies down to 0.10 TeV, thus suggesting that the spectral peak is correlated with the source luminosity. These observed characteristics could be accommodated in a synchrotron self-Compton framework in which the increase in γ-ray flux is produced by a freshly injected (high energy) electron population.