This one goes out to Dac.

Venn diagrams are meant to be graphical representations of all possible relationships between finite sets. Most often the representations are circles, though some trickier sets might require strange shapes to be used to get the point across. The diagrams can be used to represent relationships between almost any groups, but for this quick demonstration we're going to stick with using the circles to represent groups of people, since that will make it a little simpler to understand. We're just going to do a short, simple explanation of how these things work in order to improve overall understanding. So strap in PS. All diagrams were created by me, which will probably be obvious.

We'll start off as simple as we can get, in fact, I don't think this really even counts as a Venn diagram, because no relationships are being represented.
graph1.jpg
In a venn diagram, the circles represent a finite group. In this case, the circle represents all of the people who like bunnies. People on the inside of the circle, people in blue, are people who like bunnies. SImple, yeah?

Moving on, here is our second diagram:
graph2.jpg
One circle represents people who like pie. The other represents people who like cake. But wait, this diagram is not correct. Because some people like both pie and cake.

So let's see the correct diagram:
graph3.jpg
So now we see a more accurate representation (though probably not with the correct ratios). Some people, the people in the blue area, like cake but do not like pie. Other people, the people in the yellow area, like pie but do not like cake. Meanwhile some people, the people in the green area, like both cake and pie. As such, they are still inside of both circles, they still fall into both categories, but they create their own third shape in the middle, a sort of oval that represents just them, excluding the people in the blue and yellow areas. 2 circles, 3 distinct shapes and colors. So 3 distinct groups being described. Here's where people can begin to become lose what's happening. The diagram in showing the relationship between 2 groups of people is actually breaking them into 5 distinctive groups. The diagram described 5 groups of people with 5 different shapes.
1. The circle on the left describes a group where the people like cake.
2. The circle on the right describes a group where the people like pie.
3. The blue area, which is mostly a circle but has a little piece missing describes a group that likes cake but does not like pie.
4. The green area, which is sort of an oval, describes a group that likes both
5. The yellow area, once again, circle with a piece missing, describes a group that likes pie but does not like cake.
This point is essential to understanding the nature of a Venn diagram in relation to a group of people. It's describing 2 groups and their relationship to one another, but it's also breaking them into 5 distinct groups, each represented by a different shape.

Sometimes there isn't much overlap between two groups and you get a diagram like that one. Sometimes there is a lot of overlap and you get a diagram like this one:
graph4.jpg
You see, the people in one circle represent the criminals in Gotham. The people in the other circle represent people who get their asses kicked by Batman. Now, of course, some criminals in Gotham don't get beat up by Batman. These fall into the blue area. And sometimes Batman leaves Gotham to beat up bad guys elsewhere, so these people are represented by the yellow area, but most of the people Batman beats up are in Gotham and most of the criminals in Gotham get beaten by Batman at some point. So they're represented by the green area, which, in this case, is most of the diagram. Still 5 groups shown.

Now we add a third circle. We'll start with a very simple one and get into more complex relationships afterward:
graph5.jpg
So once again we have a simple description of the relationship between 2 groups. Some people love Star Wars. Some people's favorite Star Wars movie is The Empire Strikes Back, and some people fit into both groups and those people are represented by the green area. Meanwhile since the red circle represents people who have never seen Star Wars, those people can't love the series or have a favorite movie in the series. So the red circle does not overlap at all. So now we've moved up to 6 groups being shown in the diagram, but the number of relationships being described has increased to 3 or 4, because it's not just about what is overlapping anymore. It's about what is not overlapping. The red area not touching the blue, yellow, or green shows an absence of interaction just like the overlap between the blue and yellow shows the presence of an interaction.

Now we get into more complicated relationships (this whole diagram is a bit of an oversimplification, but I think you should still get the idea, just go with it):
graph6.jpg
Keep in mind what I said before that the circles and where they do not intersect is as important as where they do. You'll notice in this diagram the blue and red circles never overlap outside of the yellow circle. Because you can't be playing Halo at home alone and still be playing with other people unless you're connected to xbox live. By now you should be pretty clear on the relationship between two groups and how it describes a separate group. Like the blue and the yellow overlap to make the green, where the person is playing at home alone but is still connected to XBL. The relationship between them is described by the overlap of the blue and yellow, but [cont. in comments]