Depending on the model / brand of the laptop here are a few common things to look at:
HP DV6000 Series (and a few others): the graphics chips would heat up so much that they would un-seat themselves from the board and make the machine keep re-starting without seeing any boot screen.
Dell Machines: Generally its the inverter board that goes (as steve said), depending on the model this may be in the screen or it may be in the body of the laptop.
Apple (older): Many of the older apple machines it was also the inverter board that went, and a great many of them were in the laptop body, and required a FULL tear down to actually get to it... a very long process (see Apple Powerbook G4 Titanium)
Above is Just an example, if you can post more details, I'd be happy to see what I could find for info for you.