Three main approaches to such systems
1) Closed system liquid cooling
2) Open system liquid cooling
3) Phase change cooling. For computers some prefer to call it another thing, anybody else dealing in cooling for other things would probably laugh at the notion of separating it.
Open was favoured in the early days but has since dropped in popularity, mainly as it you have to maintain it (no leaks, possibly pull out air bubbles, top up cooling reservoir) and no kid wants to do that. Theoretically it would last longer and being modular in nature you stand a small chance of being able to replace components which have failed rather than junking the whole system or making friends with a machinist.
Closed tends to involve buying in a complete system, maybe with some rubber hoses connecting things around and installing it much like a normal fan system. It tends not to last as long as a well maintained version of the other (usually nice pinholes in your radiator) and in many cases can be nice and custom enough -- my nice fan breaks tomorrow (I say having pretty much exclusively used stock fans for years now) and I go down the local rip off merchants and live with the lawnmower on my desk for a few days while things get shipped.
Phase change is one of two things. It makes use of the idea of latent heat and works either similar to your fridge (condenses and then runs some kind of nasty refrigerant through the system which then evaporates and cools things in the process) or uses some static wax type substances that variously melt or go through glass/wax temperatures and then onto something else (or maybe that is the sole thing). Said nasty refrigerant does not have to be one of the various classical gases for it and can also be liquid nitrogen, though like most things that are not equipment costing more than your house/you earn in a few years it is mostly because it looks cool, though you can get some nice numbers on the overclocking front.
You can go further and look into things like Peltier cooling. They are not efficient but they are quiet and lack moving parts which makes them very appealing for certain uses. You can also get very silly and look at things like immersion cooling but frankly most of that is because it looks cool on your computer website/forum rather than anything terribly practical.
In most cases it is a noise and looks cool thing rather than any major cooling capacity. Industrially they can do far better but this is not that. Equally if you have a certain use cases -- it gets tedious to have to blow out computers deep in a factory or workshop so if I can seal it and use cooling like water cooling and dodge that* then that is nice. Noise is not a given though; leaving aside other fans in the case (not blowing something over your hard drives? Or are you a monster) then many liquid cooling systems will have a nice big fan to cool the radiator.
*strictly speaking I could filter the input and try for so called positive pressure inside the case but that is a pain too, not to mention nobody would ever clean the filter.
If it sounds like I am jaded and cynical it is because I am, cooling enthusiasts are not as bad as audio enthusiasts that have gone off the deep end as you at least have some objective measurements to fall back on but their approaches to the science are not good, and most of it is because it looks cool on industrial systems, just like people bolt on crap to their car because it looks cool on racing cars. There are some perks to doing it as a normal person and it is cheap enough to be a bit of fun but it is not some hidden magic field that only the best know of.
I so want to build a custom liquid cooling rig for my laptop (because it's aging and all I use it for is at the desk), but I don't know where the hell to start...
I don't think there are any custom builders that will do something like that for sensible money. The short version is remove the back, cut a hole in the shell if you prefer in the end, find the location of either the head sources or the heat sinks (they can be quite far apart on laptops), maybe bypass any fan sensors or temperature sensors, stick on a heatsink connected to said liquid cooling (buy in a system, though you might want to look into HTPC stuff to get the size down) and hook it up. You may want one of those laptop cooling pads or docking stations to raise it up a big and gain some space. Frankly though if you ripped the back of the laptop and got a nice heatsink on there instead you would probably gain a lot for your trouble.
