In my peregrinations around the Internet, I have discovered not one, but two websites devoted to the Flat Earth Society:
I think one is a splinter group from the other. I bet they hate each other.
But why limit ourselves to a binary choice? Why does the alternative to a round Earth have to be a flat Earth? Let's be creative and think of other possibilities.
Maybe the Earth is actually shaped like a torus (a gigantic donut):
This is even better than a sphere -- now there are two different ways to go around the world! You can sail around the outside of the donut, or you can circle the world through the donut hole.
Or maybe the surface of the Earth has negative curvature, like a saddle or a Pringles potato chip:
Of course, this isn't what the Earth looks like. But it's what the entire universe might look like.
The big difference between the surface of the Earth and the universe is that the Earth's surface is two-dimensional, while the universe is 3-D. So we can't really visualize the three dimensional nature of the universe -- we have to argue in analogy with the kinds of shapes that are familiar from two dimensions.
The simplest model of the universe, and the one that most cosmologists would argue for, is an infinite "flat universe" (hence the title of this post). This is ordinary three-dimensional space, going on forever in all directions. But space doesn't have to be infinite -- it's possible that space wraps back on itself if you go far enough in one direction. This is the three dimensional version of our donut. Don't try to visualize it. Your head will explode.
It's also possible that the universe is slightly curved. Observations of the cosmic microwave background (the radiation left over from the big bang) show that this curvature has to be exceedingly small, which is why many of us think it's really zero. A universe with positive curvature would look like the 3-D analog of a sphere, while negative curvature gives something like a Pringles (but in 3 dimensions). We can describe these ideas very precisely mathematically, but we just aren't built to visualize higher dimensions.
Of course, the idea that people used to believe that the earth was flat is a famous myth -- educated people have known that the world is round since ancient times. It's the universe as a whole that remains a mystery.