What is a manifold on a Euclidean space?

So the OP asked this:

if someone who doesn't study math ask you "what this a manifold?" how would you answer, in simple terms (as Feynman says).

My emphasis. So clearly the interest is about informing people who have no practice with math and requires analogies and explanations that do not include jargon. Which is great, because I am not a mathematician so I will provide the answer that was provided to me when I had this same questions hovering in my head.

You can start by saying that, the simplest example of a manifold is surface that is, in general, globally curved. As an example you can give a sphere. The surface of a sphere. Then explain that despite the fact that the sphere is curved, you can divide it in several little squares or triangles and if your division is fine enough and each little triangle is small enough, you can tile the entire surface of the sphere very tightly even though its surface is curved and the small polygonal tiles are flat. So it is locally flat, despite being globally curved. This, incidentally, is why the surface of the Earth seems so flat when seen from our vantage point.

Here you make your point that if you can tile a surface with such minuscule flat polygons, then it is a manifold because even though it is generally curved, it is locally flat. As a general example you can give a waving flag, relief terrain maps etc. 2D manifolds for which global curvature isn't constant, the most general examples you can think of.

The next point in the instruction is harder because then you need to show your interlocutor that the notion is general. That is, there is no reason why manifolds should be 2D, tiled by flat polygons. They may very well be 3D, tiled by flat polygons or 4D, tiled by flat polytopes.

Once you do that you can say that an entire manifold can, in fact, be flat. In which case it will be both globally and locally flat. When you study a manifold embedded inside a flat Euclidean space it is just a fancy way of saying that you, as the observer, is watching your curved study manifold from an external flat manifold. For instance, when you study the curvature of the surface of a sphere, a 2D manifold, inserted in a 3D flat space like the room where you are studying it, which is a 3D flat manifold. You can contrast this type of study with a study of the surface of the sphere where the observer isn't in an external room, but constrained to walk on the surface of the sphere itself.

Again, the notion of observing a sphere inside a room can be extended to the general case of $n-1$ dimensional manifolds embedded in $n$ dimensional flat manifolds.

I will leave it as homework how to explain what you mean by "Euclidean space".

I hope this helps, good luck.

We are all familiar with a line or a plane in $\mathbb R^n$.

Perhaps we are also familiar with the generalization to arbitrary dimension $k$ with $1 \le k \le n-1$ (the cases $k=0$ and $k=n$ could also be included, but for intuition's sake I'll leave them out of this discussion). Let's refer to this as a "flat $k$-dimensional subspace" in $\mathbb R^n$ (a more mathematical term is an "affine $k$-space").

At the most intuitive level one can now say:

A $k$-dimensional manifold in $\mathbb R^n$ is a subset of $\mathbb R^n$ which looks, locally, like a flat $k$-dimensional subspace in $\mathbb R^n$.

And I might go on to give examples to shore up the intuition, for example a sphere in $\mathbb R^3$.

To formalize this one further step but still keep some intuition, one can use the idea of graphs. A 1-manifold in $\mathbb R^2$ is a subset which looks, locally, like a line. But, we might also want to capture the intuition that the line can be curved. To do this, we add one more layer of formality, which might not be understandable to someone who studies no math, but would at least be understandable to an undergraduate who has had some multivariable calculus. Here's one special case:

A 1-manifold in $\mathbb R^2$ is a subset which looks, locally, like the graph of a smooth function, having one of two forms: $y=f(x)$; or $x=g(y)$.

And, more generally,

A $k$-manifold in $\mathbb R^n$ is a subset which looks, locally, like the graph of a smooth function, having one of several possible forms: for example, $(x_{k+1},...,x_n) = f(x_1,...,x_k)$; or more generally of the form $(x_{i(k+1)},...,x_{i(n)}) = f(x_{i(1)},...,x_{i(k)})$, where $i : \{1,...,n\} \to \{1,...,n\}$ is a permutation.

Again, to shore up the intuition I might write down some formulas for the sphere in $\mathbb R^3$, such as $z = \sqrt{1-x^2-y^2}$ for the upper half-hemisphere and five other similar formulas to cover the lower, left, right, front, and back hemispheres.

Only in the next step of formality would I bring in open subsets in order to formalize the meaning of the phrase "looks, locally". I wouldn't expect someone who studies no math to follow at this point, nor even an undergrad calc student, but at least this could make sense to an undergraduate math major. And at this stage the definition would no longer be just intuitive, instead it becomes the complete definition:

A subset $M \subset \mathbb R^n$ is a $k$-dimensional manifold if for each $p \in M$ there exists smooth function of the form $(x_{i(k+1)},...,x_{i(n)}) = f(x_{i(1)},...,x_{i(k)})$, where $i : \{1,...,n\} \to \{1,...,n\}$ is a permutation, and there exist open subsets $U \subset \mathbb R^n$ and $V$ of $\{(x_{i(1)},...,x_{i(k)})\}$, such that $U \cap M$ is equal to the graph of $f$.

In layman's terms, it's a space that at least for a point and it's immediate neighbors behaves like a Euclidean space (which is a set of n-tuples $\mathbb{R}^n$ equipped with a dot product, or if you're more comfortable with Physics, at least for a 3 dimensional space, is like the 3d space you use for Newtonian mechanics).

More specifically, a Manifold is a Topological space (a set of points and their neighbors satisfying some axioms), such that each point has a neighborhood of points that can be mapped with a continuous and invertible function (Homeomorphism) to a Euclidean space.

An example of what this last paragraph looks like graphically: Manifold mapping to Euclidean space example

To answer your question: no, it's not an Homeomorphic function. As I've written above, it is a space satisfying some particular properties.