We present the results of both laboratory and on sky astrometric characterization of the Gemini Planet Imager (GPI). This characterization includes measurement of the pixel scale* of the integral field spectrograph (IFS), the position of the detector with respect to north, and optical distortion. Two of these three quantities (pixel scale and distortion) were measured in the laboratory using two transparent grids of spots, one with a square pattern and the other with a random pattern. The pixel scale in the laboratory was also estimate using small movements of the artificial star unit (ASU) in the GPI adaptive optics system. On sky, the pixel scale and the north angle are determined using a number of known binary or multiple systems and Solar System objects, a subsample of which had concurrent measurements at Keck Observatory. Our current estimate of the GPI pixel scale is 14.14 +/- 0.01 millarcseconds/ pixel, and the north angle is -1.00 +/- 0.03 degrees. Distortion is shown to be small, with an average positional residual of 0.26 pixels over the field of view, and is corrected using a 5th order polynomial. We also present results from Monte Carlo simulations of the GPI Exoplanet Survey (GPIES) assuming GPI achieves similar to 1 milliarcsecond relative astrometric precision. We find that with this precision, we will be able to constrain the eccentricities of all detected planets, and possibly determine the underlying eccentricity distribution of widely separated Jovians.