I'm reading through Riemannian Geometry from DoCarmo. Chapter 0 section 2 defines what a differentiable manifold is:
A differentiable manifold of dimension $n$ is a set $M$ and a family of injective mappings $x_\alpha : U_\alpha \subset \mathbb{R}^n \to M$ of open sets $U_\alpha$ of $\mathbb{R}^n$ such that:
- $\bigcup_\alpha x_\alpha (U_\alpha) = M$
- for any pair $\alpha,\beta$ with $x_\alpha (U_\alpha) \cap x_\beta (U_\beta) = W \neq \emptyset$, the sets $x^{-1}_\alpha (W)$ and $x^{-1}_\beta (W)$ are open sets in $\mathbb{R}^n$ and the mappings $x_\beta^{-1} \circ x_\alpha$ are differentiable
- The family $\left\{ (x_\alpha,U_\alpha) \right\}$ is maximal relative to the conditions (1) and (2).
What is an example of differentiable where $M$ isn't a subset of $\mathbb{R}^n$?
Any smooth $m$-dimensional manifold $M^m$ can be smoothly embedded into $\mathbb{R}^{2m}$. So, the question would really be what $M^m$ can't be embedded into $\mathbb{R}^d$ where $d < 2m$. An example of this would be real projective space of dimension $m$. For more on this, you could look into the Whitney embedding theorem. The Nash embedding theorem may also be of interest.