Yes, there's a conversion process - involving a LOT of digital processing. Basically the incoming video signal is digitised and stored in RAM, it's then read out at a different rate, to match the computer monitor.
As well as this, it has to convert the number of lines per picture, and the number of pictures per second - a PAL television has 625 lines and shows 25 pictures per second, the line frequency is 15,625Hz, PC monitors use many times higher than that.
A standard LCD monitor is just that - A MONITOR - a laptop screen is just a screen, it doesn't include all the extra parts that make a monitor, they are integrated in the motherboard (and probably don't actually have all the sections to make a full monitor, as it's not needed).