The basics are very simple indeed. Here's how to construct European 4-in-1 maille.
You can either start with a spool of wire and construct a mandrel and cut your own rings or you can buy jump rings like I did (not having the space for such equipment, and I'm working in stainless steel to complicate things.) Sites I can recommend for supplies would be Armchair Armoury in the UK and The Ring Lord in the US/Canada.
Start with a pile of jump rings:
Taking two pairs of flat-nosed pliers, clamp them down on either side of the ring and twist to open some of the rings. I find doing strips of 10s works well for a larger project but you can do whatever length of strip you wish:
Once you have 10 open rings, get some more untouched rings and then close 22:
Put 4 closed rings through 1 open ring and then close that ring. That will serve as your starting point. Then put 2 closed rings over the other 9 open rings but do not close them just yet:
Lay that starting point out like so:
Notice that the middle ring has 4 rings though it. This is what we mean by European 4-in-1 maille, every ring should be joined to 4 others. There are many other kinds easily findable via web search, such as dragonscale maille.
Next, take one of those still-open rings with 2 closed rings on it and thread the open ends through the starting point's closed rings like so:
Flip the whole thing over and use your pliers to twist closed those open ends on the other side, then lay it back out as before.:
Rinse and repeat and soon you will have a strip of 10:
If you lay two strips side by side you will see four closed rings just ready to have an open ring threaded through (from right to left in this case, through the rightmost column's middle two rings, and then the second column's) and then the ends closed using pliers.