First and Second Fundamental Form Intuition

Intuitively, the first fundamental form tells you how to compute the distances along the paths within the surface (it is just a Riemannian metric of the surface thought as a standalone manifold, that is if we forget about the embedding/immersion). This explains why it is also called the intrinsic metric.

The second fundamental form describes how "curved" the embedding is, in other words, how the surface is located in the ambient space. It is a kind of derivative of the unit normal along the surface or, equivalently, the rate of change of the tangent planes, taken in various directions within the surface. Alternatively, it is called the shape tensor (it has a close relation to the shape, or Weingarten, operator), and is an extrinsic quantity in the sense that it depends on the embedding.

The Bonnet theorem (see a discussion here) ensures that (under certain conditions) these two fundamental form uniquely characterize the surface (locally), that is we can "integrate" them to a piece of surface in the space uniquely up to a rigid motion of the space.

The bottom line is that the Ist and IInd fundamental forms are as good as a complete set of local invariants of a surface, and thus they are extremely useful and important in differential geometry.

Remark 1. With regards to the coefficients, the comments have fully addressed this question: they are just components of these tensors in a coordinate patch.

Remark 2. The Christoffel symbols is a coordinate way to represent the invariant differentiation of vector (and all tensor) fields along the surface that arises from the given structures. In our case we have the usual (standard, Euclidean) metric in the ambient space and the Levi-Civita connection of this metric is just the usual (flat, Euclidean) derivative (just partial derivatives of the component in the standard coordinates). This (ambient) connection has its own Christoffel symbols but in our setting they all are zero, so it is customary not to mention them. Taking a vector field tangential to the surface we can try to differentiate it with this ambient derivative but for this to work we need to extend this vector field off the surface. The result of the differentiation will certainly depend on the extension but the tangential part of this result turns out to be independent of extensions when restricted to the surface. This way we obtain the covariant derivative (of tangential vector, tensor, ... fields) in the surface, and the Christoffel symbols that you may have met are the "components" of this covariant derivative (the Levi-Civita connection of the first fundamental form).

Let $\mathbf{r}$ be a vector in the containing space, e.g., $(x,y,z)$ in three dimensions. Let $\mathbf{w}$ be coordinates in the embedded manifold, e.g., $(u,v)$ in a two-dimensional manifold. Assume that components of $\mathbf{r}$ can be written as functions of $\mathbf{w}$, and vice versa. Then the metric tensor is defined as

$g_{ij} = \sum_k \frac{dr^k}{dw^i} \frac{dr^k}{dw^j}$

where letter superscripts are indices, not powers. $g$ is clearly symmetric.

The square of the arc length is $ds^2 = \sum_{ij} g_{ij} dw^i dw^j$.

In two dimensions, $ds^2 = g_{11} du^2 + g_{12} du dv + g_{21} dv du + g_{22} dv^2$, or
$E du^2 + 2 F du dv + G dv^2$. This is the first fundamental form for a surface.

The components of the shape tensor are projections of second partial derivatives onto the unit normal. In two dimensions, a unit normal is the cross product of the tangent vectors, which are derivatives of $\mathbf{r}$ with respect to $u$ and $v$. The shape tensor is given by

$b_{ij} = \sum_k n^k \frac{\partial^2 r^k}{\partial w^i \partial w^j}$,

which is clearly symmetric. The distance from the surface at r+dr to the tangent plane at r is given by

$2D = \sum_{ij} b_{ij} dw^i dw^j$.

In two dimensions this is $2D = b_{11} du^2 + b_{12} du dv + b_{21} dv du + b_{22} dv^2,$ or $2D = e du^2 + 2 f du dv + g dv^2$. This is the second fundamental form for a surface.

A good reference is sections 32-36 of Vector and Tensor Analysis by Harry Lass, McGraw-Hill (1950).