Why is DC motor speed and torque direction the same, whereas in a generator it is the opposite?
That's two separate questions. From the title:


  Why in dc motor speed and torque direction is same where as in generator is opposite?


Because the flow of mechanical power (into or out of the machine) is opposite in the two cases.


  Whats the relation between speed and torque in dc machine i know its inversely proportional but how?? Torque is nothing but twisting moment right.. if torque increases then speed must be increased but its wrong why??


Power is speed &times; torque. Therefore if power is constant, then the two factors must be inversely proportional. If they both increase, then the power must increase, too.