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Resistivity and magnetization have been measured at different temperatures and magnetic fields in organic superconductors $\kappa$-(BEDT-TTF)$_{2}$Cu[N(CN)$_{2}$]Br. The lower critical field and upper critical field are determined, which allow to depict a complete phase diagram. Through the comparison between the upper critical fields with magnetic field perpendicular and parallel to the conducting ac-planes, and the scaling of the in-plane resistivity with field along different directions, we found that the anisotropy $\mathit{\Gamma}$ is strongly temperature dependent. It is found that $\mathit{\Gamma}$ is quite large (above 20) near $T_{\rm c}$, which satisfies the 2D model, but approaches a small value in the low-temperature region. The 2D-Tinkham model can also be used to fit the data at high temperatures. This is explained as a crossover from the orbital depairing mechanism in high-temperature and low-field region to the paramagnetic depairing mechanism in the high-field and low-temperature region. The temperature dependence of lower critical field $H_{\rm c1} (T)$ shows a concave shape in wide temperature region. It is found that neither a single $d$-wave nor a single $s$-wave gap can fit the $H_{\rm c1} (T)$, however a two-gap model containing an $s$-wave and a $d$-wave can fit the data rather well, suggesting two-band superconductivity and an unconventional pairing mechanism in this organic superconductor.