What is the difference between a manifold and a topological manifold? Is it the case that a topological manifold is just a topological space but we have not mentioned any specific structure on it ?
 A: What is ment by a "manifold" is usually smooth manifold. Specifically, it is a topological manifold such that the transition maps are all smooth.
That is, a manifold is a topological space $X$ such that it has an open cover $\{U_i\}$ with homeomorphisms $x_i:U_i \to \mathbb R^n$, and such that on each intersection $U_i \cap U_j$, the maps $x_j \circ x_i^{-1}:\mathbb R^n \to \mathbb R^n$ are all smooth maps. 
Notice that this is a lot stronger than just requiring the space $X$ to be locally Euclidean.
Also, there are examples of homeomorphic topological manifolds that are not isomorphic as smooth manifolds. Search for "Milnor exotic spheres" for example. Or search for the uncountable number of smooth structures on $\mathbb R^4$. 
So yes: a topological manifold is just a space $X$ that is locally euclidean. You also usually require it to be Hausdorff (so as to exclude the line with two origins, for example). 
A: A topological manifold $M$ is a topological space that's locally homeomorphic to a subset of $\mathbb R^n$: for every point $p \in M$, there is an open neighborhood $p \in U \subset M$ such that $U$ is homeomorphic to a subset of $\mathbb R^n$.
A smooth manifold (or piecewise linear manifold, etc...) is a topological manifold $M$ with a smooth (resp. PL) structure: an open cover $\{U_\alpha, \varphi_\alpha\}$ of $M$ of sets homeomorphic to subsets of $\mathbb R^n$ (along with these homeomorphisms) such that $$\varphi_\beta \circ \varphi^{-1}_\alpha: \varphi_\alpha(U_\beta \cap U_\alpha) \rightarrow \varphi_\beta(U_\beta \cap U_\alpha)$$ is smooth (resp. piecewise linear) for any $\alpha, \beta$. Note that $\varphi_\alpha(U_\alpha)$ is a subset of $\mathbb R^n$, where we already have notions of smooth or piecewise linear.
So a topological manifold is a topological space with a certain pleasant property. A smooth manifold (or PL manifold, or complex manifold, or affine manifold, or...) is a topological manifold with a smooth (PL/complex/affine...) structure.
There is definitely a difference between these two concepts: a topological manifold can support multiple non-diffeomorphic smooth structures, and some topological manifolds do not support a smooth structure at all.
