European Four in One is the most basic maille weave and a common choice for beginning maillers. It can be used for both armor and jewelery; it was the most common weave for medieval European armor. It works well in an incredibly wide range of ring sizes, from 1/2" 14 gauge (good costume maille in steel) to 1/8" 20 gauge (for delicate jewelery). The minimum ring size for some common gauges are: 14 gauge, 1/4", 16 gauge, 3/16", 18 gauge, 5/32". At these ring sizes, the weave will be very thick and rather inflexible, making for heavy armor or solid jewelery. If you're planning on making armor, then 14 gauge 3/8" steel rings or 12.5 gauge 3/8" aluminum rings will work well.

The basic technique for making European Four in One is to make units, weave the units into chains, and then weave the chains into sheets. For European Four in One, a unit is a single ring with four other rings hooked into it, and it looks like this:

You'll need a lot of these units when you weave Euro 4-1, so you might as well make several now. Once you have a few units made, you'll need to connect them into a chain. To do this, pass a new ring through two of the rings on one unit and two of the rings on the other unit, like so:

Professional armorers generally make huge lengths of this "Unit Chain," often winding them onto spools so that they can easily cut off what they need for a given application. You don't need to make that much.
Now, there is a certain rule to keep in mind about unit chains. They should always begin and end with a single link, so that you can count the length of a chain by counting the number of single links. This allows for easy communication; if you need a unit chain six links long, for example...

...then you'll know exactly what you should make. Without this rule, people tend to get confused trying to count units or individual rings. Another benefit of using unit chains this way is that when you weave two chains together, they will always match up. If you ignore the rule, then occasionally you'll have to add a single row of links so that you can connect two chains, and nobody wants to do that.
So, to connect two unit chains, lay them out as shown:

Pass a ring through one ring on one chain and one ring on another:

And continue adding rings on down the chains. Excepting the rings on the end, each ring that you add will pass through four rings in total - two on one chain, two on the other.

Congratulations! You now have a sheet of European Four in One, and that makes you an official maillesmith! Have fun! But first, we should probably make an observation on the units with which you've been working. You might have noticed that there's not just one way to connect units; we can do it in the other direction as well:

This is called a "Ribbon Chain," and it can be expanded into sheets just like unit chains can be. Line up two ribbon chains side by side, like so:

And connect them down the middle.

So why don't we use this method instead? Well, there are a few reasons. First off, it prevents us from using that handy rule that we can use for unit chains. This can lead to all kinds of confusion when two maillesmiths try to communicate. Secondly, if we use ribbon chains instead of unit chains, we may occasionally find ourselves having to add a single row of links to the weave just to get things to match up. Finally, it's considerably more difficult to weave ribbons than it is to weave units.
However, nobody's perfect, and you may well find yourself needing an extra column of maille here or there. In those cases, ribbon chains can be useful. They just aren't recommended for general-purpose weaving.

Now that you've finished the first weave of maille, you might be interested in trying out a different weave. From here, I recommend either box chain or byzantine. Both make good jewelery and are relatively simple. Good luck!