"A manifold is a topological space which is locally Euclidean ..." source: http://mathworld.wolfram.com/Manifold.html
"locally Euclidean" simply means that on short distance scales ("local") it exhibits the topological characteristics of flat Euclidean Geometry.
A topological space is the most primitive (least complex) mathematical structure:
In the chapter "Point Sets in General Spaces" Hausdorff (1914) defined his concept of a topological space based on the four Hausdorff Axioms.1. To each point x there corresponds at least one neighborhood U(x), and U(x) contains x.
2. If U(x) and V(x) are neighborhoods of the same point x, then there exists a neighborhood W(x) of x such that W(x) is a subset of the intersection of U(x) and V(x).
3. If y is a point in U(x), then there exists a neighborhood U(y) of y such that U(y) is a subset of U(x).
4. For distinct points x and y, there exist two disjoint neighborhoods U(x) and U(y).
source: http://mathworld.wolfram.com/TopologicalSpace.html
I'm afraid there's no good way to state this in laymans' terms without losing the precision of the axioms.