We present the Early Data Release of the Sydney-AAO Multi-object Integral field spectrograph (SAMI) Galaxy Survey. The SAMI Galaxy Survey is an ongoing integral field spectroscopic survey of ~3400 low-redshift (z<0.12) galaxies, covering galaxies in the field and in groups within the Galaxy And Mass Assembly (GAMA) survey regions, and a sample of galaxies in clusters. In the Early Data Release, we publicly release the fully calibrated data cubes for a representative selection of 107 galaxies drawn from the GAMA regions, along with information about these galaxies from the GAMA catalogues. All data cubes for the Early Data Release galaxies can be downloaded individually or as a set from the SAMI Galaxy Survey website. In this paper we also assess the quality of the pipeline used to reduce the SAMI data, giving metrics that quantify its performance at all stages in processing the raw data into calibrated data cubes. The pipeline gives excellent results throughout, with typical sky subtraction residuals in the continuum of 0.9-1.2 per cent, a relative flux calibration uncertainty of 4.1 per cent (systematic) plus 4.3 per cent (statistical), and atmospheric dispersion removed with an accuracy of 0.09-arcsec, less than a fifth of a spaxel.