We could begin with a simple definition of a manifold, the informal definition we find on wikipdia, space that is "modeled on" Euclidean space." Even less formally, space that is based on square, straight space.
Why do this? Well, we know that Euclidean space is just one space, and mathematicians needed a formal way to discuss this, and to find a way to deal with how humans think about things on earth, which is, mostly treated as a Euclidean space, even though it is not. So we call this a manifold to keep in mind the reality that, on the scale of our lives, things are approximately Euclidean, but not really on the larger scale. Thus, the sphere* on which we live is a 2-manifold, since, in a local area, it approximates a 2-dimensional Euclidean space (not taking into account the vertical dimension, nor the time-like dimension).
This allows us to do useful things, like measure distances in units that we recognize, or make maps (!) and atlases. The non-Euclidean nature of our world becomes evident when we try to make a map that covers a large region. Cartographers then employ several methods to make maps that will work for one use or another, but are never quite right over the entire surface.