if it works ok on external monitor it may be the screen inverter--look very closely at the screen if the inverter has gone you should just be able to make out the desktop even though the screen is dark
or it may be the ribbon connecting the motherboard to the display
both of these are common faults and quite cheap to repair if you have the ability to do it yourself--i usually replace the inverter 1st as its easier to get to
most laptops also can toggle between internal and external screens so check that as well in case --usually done by pressing the fn key and another key together--which other key depends on the model you have--hp and ibm for example its fn and f5 or fn and f4
the last possibilty is the ccl bulb on the screen has gone which usually means replacing the screen--this is far less common