The Andromeda galaxy (M 31) is one of a few galaxies that has sufficient angular size on the sky to be resolved by the Planck satellite. Planck has detected M 31 in all of its frequency bands, and has mapped out the dust emission with the High Frequency Instrument, clearly resolving multiple spiral arms and sub-features. We examine the morphology of this long-wavelength dust emission as seen by Planck, including a study of its outermost spiral arms, and investigate the dust heating mechanism across M 31. We find that dust dominating the longer wavelength emission (greater than or similar to 0.3 mm) is heated by the diffuse stellar population (as traced by 3.6 mu m emission), with the dust dominating the shorter wavelength emission heated by a mix of the old stellar population and star-forming regions (as traced by 24 mu m emission). We also fit spectral energy distributions for individual 5' pixels and quantify the dust properties across the galaxy, taking into account these different heating mechanisms, finding that there is a linear decrease in temperature with galactocentric distance for dust heated by the old stellar population, as would be expected, with temperatures ranging from around 22 K in the nucleus to 14 K outside of the 10 kpc ring. Finally, we measure the integrated spectrum of the whole galaxy, which we find to be well-fitted with a global dust temperature of (18.2 +/- 1.0) K with a spectral index of 1.62 +/- 0.11 (assuming a single modified blackbody), and a significant amount of free-free emission at intermediate frequencies of 20-60 GHz, which corresponds to a star formation rate of around 0.12 M-circle dot yr(-1). We find a 2.3 sigma detection of the presence of spinning dust emission, with a 30 GHz amplitude of 0.7 +/- 0.3 Jy, which is in line with expectations from our Galaxy.