We present a structural and morphological catalogue for 45 million objects selected from the first year data of the Dark Energy Survey (DES). Single Sersic fits and non-parametric measurements are produced for g, r, and i filters. The parameters from the best-fitting Sersic model (total magnitude, half-light radius, Sersic index, axis ratio, and position angle) are measured with GALFIT; the non-parametric coefficients (concentration, asymmetry, dumpiness, Gini, M20) are provided using the Zurich Estimator of Structural Types (ZEST+). To study the statistical uncertainties, we consider a sample of state-of-the-art image simulations with a realistic distribution in the input parameter space and then process and analyse them as we do with real data: this enables us to quantify the observational biases due to PSF blurring and magnitude effects and correct the measurements as a function of magnitude, galaxy size, Sersic index (concentration for the analysis of the non-parametric measurements) and ellipticity. We present the largest structural catalogue to date: we find that accurate and complete measurements for all the structural parameters are typically obtained for galaxies with SEXTRACTOR MAG_AUTO_I <= 21. Indeed, the parameters in the filters i and r can be overall well recovered up to MAG_AUTO <= 21.5, corresponding to a fitting completeness of similar to 90 per cent below this threshold, for a total of 25 million galaxies. The combination of parametric and non-parametric structural measurements makes this catalogue an important instrument to explore and understand how galaxies form and evolve. The catalogue described in this paper will be publicly released alongside the DES collaboration Y1 cosmology data products at the following URL: https://des.ncsa.illinois.edu/releases.