A topological manifold is a topological space which locally resembles real $n$-dimensional Euclidean space. Here I consider removing `Euclidean' from manifold:
Suppose that $X$ is a topological space, and $M$ is a topological space which locally resembles $X$, i.e. for each $p\in M$, there is an open neighborhood $U$ of $p$ in $M$, an open set $V$ in $X$, and a homeomorphism $\varphi: U\rightarrow V$. Is there a name for $M$?
And is there a book about orbifolds for beginners? I only know some basic concepts of differential manifolds.
As far as I know, there is no name for such spaces. One just call them locally homeomorphic to the space $M$. Usually one needs to relate with the local homeomorphism the topological space and a euclidean space, since the euclidean structure coupled with the local homeomorphisms allows us to bring all differential calculus on the manifold.
Let me build on Christoph’s suggestion instead (thank you, I was not aware of them either): if the space $X$ is endowed with at least a structure of pseudogroup, then one may want to call the whole manifold a $G$-manifold.
For a fast introduction to orbifolds, I would give a first look here.