You don't say which type of monitor. This question gets asked so many times on here that it should be in the sticky.
Basically you will need a VGA box of some description. These range from about £40 to a few hundred. The cheap ones will allow you to connect a PS2 to your monitor via the right cable but the quality will be VERY poor, No where near TV quality and not worth it. To get a half decent picture you will need an expensive box capable of upscaling properly to your monitor's native resolution. £200 just to play a PS2? I don't think so.
A completely free option is to use a program such a DeScaler, this works via your graphics card's video-in. This again is very poor compared to tv and is a pain to set up.
Of course if you don't care about picture quality then either the cheap
VGA box or DeScaler will work.
Be warned, when I say the quality is poor, I mean VERY poor.