A "flat" universe is basically one that can be described with Euclidean geometry. In other words, it means that you can use those ordinary equations for the areas and volumes of spheres, cubes, etc. that you learned in grade school. If it were curved, then you'd find, for example, that the pythagorean theorem didn't work on triangles (though you probably wouldn't notice unless your triangle was really big).
It's impossible for humans to imagine things in 4-D, so it's best to think of the 3-D analogy. Imagine you're a 2-D person living on the surface of a balloon. Since you're a 2-D person, you can't see that third dimension in which the balloon is curved, so you probably assume that normal euclidean geometry will work to describe the size of your triangular house -- it's a right triangle, cause that's what all the upper class 2-D people live in over in two-dimensional suburbia. But one day, you go out and actually measure its sides and...*gasp*... a^2+b^2 \ne c^2.
How can this be? It's because the surface you're living on is curved and the equations that describe spherical triangles are different from those that describe flat ones. But let's say that there's another 2-D person who's living on a piece of paper over in Chicago. This person is living on a flat surface, so they will measure their right triangle house to obey the Pythagorean theorem and everything will seem normal (or, at least, as normal as possible for someone living in Chicago).
Consider one last analogy. Imagine we're back to the person on the surface of the balloon, but imagine this time that the balloon is much, MUCH larger than the 2-D person (or even their observable universe). In this case, the surface they're living on would be curved, but they wouldn't be able to tell because they would only be able to measure a very small portion of the balloon and, to the accuracy with which they can measure, that portion would appear flat (same basic reason that the Earth appears flat to someone living on it).
The reason I give the last analogy is that it describes the basic idea behind inflation. That is, the universe appears flat because we're only observing a very small portion of it. It suggests that, at some point, the "balloon" that is our universe expanded so much that the geometry in the present day observable universe is, for all intents and purposes, flat.
Keep in mind that this is just an analogy and that you have to add a dimension (specifically, a time-like one) in order for it to apply to cosmology.
