How do I run a wire underground to an outdoor post lamp?
According to NEC table 300.5, for using a branch circuit of 120V and not more than 20A, with GFCI protection:


If it's under 2" thick concrete or equivalent, then depth = 6" and you can use direct burial-rated -- i.e. type UF -- cable, or run in a conduit 
Under a driveway (for 1-2 family homes), depth = 12"
Under lawns (and anywhere except streets, buildings, parking lots), depth = 12" for direct burial-rated cable, or 6" in a metallic conduit