If $k\le d$, then $M\subseteq\mathbb R^d$ is a $k$-dimensional $C^\alpha$-submanifold iff for all $p\in M$, there is an open neighborhood $U\subseteq\mathbb R^d$ and a $f\in C^1(U,\mathbb R^{d-k}$ with $M\cap U=\{f=0\}$ and such that ${\rm D}f(x)$ has full rank for all $x\in U$. Now there is an equivalent definition which states that $M$ is a $k$-dimensional $C^\alpha$-submanifold iff for all $p\in M$, there is an open neighborhood $U\subseteq\mathbb R^d$ and a $C^\alpha$-diffeomorphism $F:U\to F(U)$ with $F(M\cap U)=V\cap(\mathbb R^k\times\{0\})$.
Now consider, for example, the unit sphere $M:=S_{d-1}=\{x\in\mathbb R^d:|x|=1\}$. Considering $$f(x):=|x|^2-1\;\;\;\text{for }x\in\mathbb R^d,$$ it's easy to see that $M$ is a $(d-1)$-dimensional $C^\infty$-submanifold and $f$ is as in the first definition.
How does the corresponding diffeomorphism from the second definition look like?