M1330 Inverter

First, verify that there is a good image on an external display.  If there is not, chances are it's the video chip that has failed - not the inverter.  That is a known problem with the nVidia 8400GS chip on many of these m1330 systems, and there's a 1-year warranty extension that covers the required mainboard replacement:

http://en.community.dell.com/blogs/direct2dell/archive/2008/08/18/nvidia-gpu-update-dell-to-offer-warranty-enhancement-to-all-affected-customers-worldwide.aspx

If there IS a good image on an external monitor, it's a tossup as to whether it's the inverter or the backlight.  Probably 30% of the time it's the inverter - 70% the backlight.  The only way to source the correct inverter is to remove the existing one to find out its model number and search for an exact replacement.  Dell used at least half a dozen screen models in the m1330, not all of which use the same inverter. 

You will find the inverter behind the screen itself - follow the two wires from the mainboard to the inverter to locate it.  Remove it and order the exact same part to replace it.

If the inverter doesn't solve the problem - and it may not - you then know you'll need a backlight replacement (which is beyond home repair for most - you'll need a pro to do it correctly).

Thank you very much for your reply.

I connected an external monitor to the laptop, and everything was displayed normally on the external monitor.

I also tried to push the "D" key and the power button simultaneously to turn the laptop on.  The screen did some kind of color test: it displayed WHITE, RED, GREEN and BLUE color in turn on the whole screen twice.  And I could see the backlight.  Does it mean that the backlight works?

If the light stayed on, yes -- the backlight and inverter are OK.  Something else - most probably the system board - is not.