We present the largest sample of flares ever compiled for a single M dwarf, the active M4 star GJ 1243. Over 6100 individual flare events, with energies ranging from 1029 to 1033 erg, are found in 11 months of 1 minute cadence data from Kepler. This sample is unique for its completeness and dynamic range. We have developed automated tools for finding flares in short-cadence Kepler light curves, and performed extensive validation and classification of the sample by eye. From this pristine sample of flares we generate a median flare template. This template shows that two exponential cooling phases are present during the white-light flare decay, providing fundamental constraints for models of flare physics. The template is also used as a basis function to decompose complex multi-peaked flares, allowing us to study the energy distribution of these events. Only a small number of flare events are not well fit by our template. We find that complex, multi-peaked flares occur in over 80% of flares with a duration of 50 minutes or greater. The underlying distribution of flare durations for events 10 minutes and longer appears to follow a broken power law. Our results support the idea that sympathetic flaring may be responsible for some complex flare events.