The concept of Euclidean space is remarkably known from high school days, but the more complicated space in the name of “manifold” is not quite so. Let us start with the simple example. The example is a 3 dimensional sphere.
On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180°. A sphere can be represented by a collection of two dimensional maps, therefore a sphere is a manifold.
 Wikipedia article, http://en.wikipedia.org/wiki/Manifold