/ AbstractWe report on the observations of 14 dwarf spheroidal galaxies with the Fermi Gamma-Ray Space Telescope taken during the first 11 months of survey mode operations. The Fermi telescope provides a new opportunity to test particle dark matter models through the expected gamma-ray emission produced by pair annihilation of weakly interacting massive particles (WIMPs). Local Group dwarf spheroidal galaxies, the largest galactic substructures predicted by the cold dark matter scenario, are attractive targets for such indirect searches for dark matter because they are nearby and among the most extreme dark matter dominated environments. No significant gamma-ray emission was detected above 100 MeV from the candidate dwarf galaxies. We determine upper limits to the gamma-ray flux assuming both power-law spectra and representative spectra from WIMP annihilation. The resulting integral flux above 100 MeV is constrained to be at a level below around 10^-9 photons cm^-2 s^-1. Using recent stellar kinematic data, the gamma-ray flux limits are combined with improved determinations of the dark matter density profile in 8 of the 14 candidate dwarfs to place limits on the pair annihilation cross-section of WIMPs in several widely studied extensions of the standard model. With the present data, we are able to rule out large parts of the parameter space where the thermal relic density is below the observed cosmological dark matter density and WIMPs (neutralinos here) are dominantly produced non-thermally, e.g. in models where supersymmetry breaking occurs via anomaly mediation. The gamma-ray limits presented here also constrain some WIMP models proposed to explain the Fermi and PAMELA e^+e^- data, including low-mass wino-like neutralinos and models with TeV masses pair-annihilating into muon-antimuon pairs. (Abridged)