Two definitions of covariant derivatives in the setting of surfaces in $\mathbb{R}^3$ 
Suppose that $\gamma$ is a curve on a surface $S\subset\mathbb{R}^3$ and let $v$ be a tangent vector field along $\gamma$, i.e., a smooth map from an open interval $(\alpha,\beta)$ to $\mathbb{R}^3$ such that $v(t)\in T_{\gamma(t)}S$ for all $t\in(\alpha,\beta)$. If $N$ is a unit normal to $\sigma$, the component of $\dot v$ perpendicular to the surface is $(\dot v\cdot N)N$, so the tangential component is
  $$
\nabla_\gamma v=\dot v-(\dot v\cdot N)N. \tag{1}
$$
  This is defined as the covariant derivative of $v$ along $\gamma$.
  

For an abstract Riemannian manifold, if one considers its embedding into an Euclidean space, then one has a definition of covariant derivatives similar to (1). This Wikipedia article calls it an "informal definition". On the other hand, there is a formal definition using connections.  

Using the example of surfaces in $\mathbb{R}^3$, could one describe the relations between the informal definition (1) with its formal one (by connections)? (How would one turn (1) into the one using connections?)
 A: The following is an important fact which basically means that Levi-Civita connections behave well under isometric embeddings. (It can actually be stated for isometric immersions as well).
Fact (or exercise, if you like): Let $\widetilde{M}$ be a Riemannian manifold, and let $\widetilde{\nabla}$ denote the Levi-Civita connection of $\widetilde{M}$. Let $M$ be a submanifold of $\widetilde{M}$, equipped with the induced Riemannian metric, and let $X,Y$ be vector fields on $M$. Let $\widetilde{X},\widetilde{Y}$ be vector fields on $\widetilde{M}$ which extend $X,Y$, respectively. Then:
i) The orthogonal projection of $\widetilde{\nabla}_{\widetilde{X}}\widetilde{Y}$ on $TM$ is independent of the extensions $\widetilde{X},\widetilde{Y}$.
ii) Let $\nabla_XY$ denote the orthogonal projection of i). Then $\nabla$ is the Levi-Civita connection of $M$.
Now, to your question. The Levi-Civita connection of $\mathbb{R}^3$ (with the standard metric) is just the usual derivative. As follows from the above fact, what described in the post is covariant derivation with respect to the Levi-Civita connection on the surface $S$.
Historical remark: Gauss and his gang knew how to take covariant derivatives on surfaces many years before Levi-Civita was born, and before the term "connection" was ever used in its current geometrical meaning. However, the approach described above is the one which is most commonly used today. It helps one to put all the pieces together.
