A (topological) manifold is a space $M$ which is: 

(1) "locally" "equivalent" to $\mathbb{R}^n$ for some $n$. 

"Locally", the "equivalence" can be expressed via $n$ coordinate functions, $c_i: M \to \mathbb{R}$, which together form a "structure-preserving" function, $c: M \to \mathbb{R}^n$, called a *chart*.
   
(2) can be realized in a "structure-preserving" way as a subset of $\mathbb{R}^N$ for some $N \ge n$. [(1)][1][(2)][2]

Note that in order to make "structure" precise here, one needs to understand basic notions of [_topology_][3], which allows one to make precise notions of ["local"][4] behavior, and thus ["locally"][4] above. When I say "equivalent", I mean equivalent topological structure ([homeomorphic][5]), and when I say "structure-preserving" I mean the same thing (creates an equivalent topological structure).

Note also that in order to [do calculus on manifolds][6], one needs an additional condition which doesn't follow from the above two conditions, which basically says something like "the charts are well-behaved enough to allow us to do calculus". These are the manifolds most often used in practice. Unlike general topological manifolds, in addition to calculus they also allow [triangulations][7], which is [very important in applications like yours involving point cloud data][8].

Note that not all people use the same definition for a (topological) manifold. Several authors will define it [as satisfying only][9] condition (1) above, not necessarily also (2). However, the definition which satisfies both (1) and (2) is much better behaved, therefore more useful for practitioners. One might expect intuitively that (1) implies (2), but it actually doesn't.


  [1]: https://en.wikipedia.org/wiki/Whitney_embedding_theorem
  [2]: https://math.stackexchange.com/questions/2176104/is-this-alternative-definition-of-manifolds-correct
  [3]: https://en.wikipedia.org/wiki/General_topology
  [4]: https://en.wikipedia.org/wiki/General_topology#Neighborhood_definition
  [5]: https://en.wikipedia.org/wiki/Homeomorphism
  [6]: https://en.wikipedia.org/wiki/Differentiable_manifold
  [7]: https://en.wikipedia.org/wiki/Triangulation_(topology)
  [8]: http://www-pequan.lip6.fr/~tierny/private/topologicalDataAnalysis.pdf
  [9]: https://math.stackexchange.com/questions/2117058/is-this-hierarchy-of-manifolds-correct