I appologize in advance in case this is a very trivial issue and for any mistakes due to translating stuff from my German lecture notes to English ...
A subset $M \subset \mathbb{R}^n$ is defined to be a $k$-dim submanifold of $\mathbb{R}^n$ with $1 \le k \le n-1$, if each point $m \in M$ has the following property:
There is an open set $U \subset \mathbb{R}^n $ surrounding $m$ and an open subset $V \subset \mathbb{R}^n$ related by a diffeomorphism $\Phi: U \rightarrow V$ such that
$$ \begin{eqnarray} \nonumber \Phi(U \cap M) & = & V \cap(\mathbb{R}^k \times \{ 0 \}) \doteqdot W \\ & = & \{ x =(x_1, ...,x_n) \in V | x = (x_1, ... ,x_k, 0, ..., 0) \} \end{eqnarray} $$
This basically means, that $M\cap U$ locally corresponds to $\mathbb{R}^k$.
A local chart of $M$ at point $m$ is given by the pair $(W,\phi \doteqdot \Phi^{-1})$
From another source, a k-dim manifold is described to be locally $\mathbb{R}^k$ too and covered by a family of local coordinate systems.
What is the difference between a manifold and a submanifold of the same dimension?
Is it just that the sumbanifold is "embeded" into a higher-dimensional space, whereas the manifold is not? To me it seems both can be covered in a very similar way by local coordinates.
An abstract manifold of dimension $k$ is defined by a family of charts (local coordinate systems). In particular, such a family of charts exists for every $k$-dimensional submanifold of $\mathbb{R}^n$.
On the other hand, manifolds may be constructed independent of a realization in a Euclidean space, e.g., by surgery. There are 2-dimensional manifolds such as the Klein bottle that cannot be realized (without self-intersection) as 2-dimensional submanifold of $\mathbb{R}^3$.
The Whitney embedding theorem says that a $k$-dimensional manifold can be diffeomorphically embedded into $\mathbb{R}^{2k}$ iff its topology is Hausdorff and second countable. Thus the concept of an abstract manifold is slightly more general than a submanifold but the collection of nice examples is the same (up to diffeomorphisms).
Note that there may be multiple differentiable structure on the same manifold considered as a topological space. Examples are exotic spheres.