Parameter estimation for two-dimensional hypoelliptic diffusions is considered within two observations frameworks: complete observations where both coordinates are discretely observed and partial observations where only the first coordinate is discretely observed. Since the volatility matrix is degenerate, Euler contrast estimators can not be used directly. For complete observations, we introduce an Euler contrast based on the second coordinate only. For partial observations, we define an Euler contrast for an integrated diffusion resulting from a transformation of the original one. We present a theoretical study where the estimators are proved to be consistent and asymptotically Gaussian. A numerical application to Langevin systems illustrates the nice properties of both complete and partial observations estimators.