The Jordan Curve Theorem (named for a French mathematician who first proved it) states that any continuous simple closed curve in a plane, separates the plane into two disjoint regions, the inside and the outside.
...Seems intuitively pretty straightforward, or as one of the books I have on my shelf says,
"The theorem seems like a statement of the blindingly obvious. If a curve proceeds continuously without any breaks in it and returns to its starting point without crossing itself, then there will be a region outside the curve and a region inside. The two regions are separate, one is finite and the other is infinite."
While this is indeed clear for the run-of-the-mill closed curves we are accustomed to seeing, there are far more complex topological curves, including for example the Koch Snowflake, that may help one see why the theorem's proof (and it has been proved) is not at all easy (some of the proofs run to 1000's of lines).
A couple of discussions of the theorem here: