First-principles calculations of the essential spin-orbit and spin relaxation properties of phosphorene are performed. Intrinsic spin-orbit coupling induces spin mixing with the probability of b(2) approximate to 10(-4), exhibiting a large anisotropy, following the anisotropic crystalline structure of phosphorene. For realistic values of the momentum relaxation times, the intrinsic (Elliott-Yafet) spin relaxation times are hundreds of picoseconds to nanoseconds. Applying a transverse electric field ( simulating gating and substrates) generates extrinsic C-2v symmetric spin-orbit fields in phosphorene, which activate the D'yakonov-Perel' mechanism for spin relaxation. It is shown that this extrinsic spin relaxation also has a strong anisotropy and can dominate over the Elliott-Yafet one for strong enough electric fields. Phosphorene on substrates can thus exhibit an interesting interplay of both spin-relaxation mechanisms, whose individual roles could be deciphered using our results.