Dog's Dick is a word mostly used to describe someone who is useless, It can also be used as a way of offending and verbaly abusing people very deeply but without saying any of the incredibly harsh words such as Cunts or Fuckers.
Although Dog's Dick doesnt really sound offensive,the user and certainly the recipient will understand its direct and degrading nature.
Fred - "Dave can u pick me up from the bus-station? Andys forgot to get me!"
Dave-"haha, ok pal...ill be there in ten,cant believe he forgot you"
Fred -"He fucks me off sumtimes,hes such a fucking Dog's Dick"