What exactly is the Metric Tensor? I have recently watched (part of) a video by DrPhysicsA on Einstein's Field Equations. Now, he starts right off by explaining the Metric Tensor is through lots of equations that made general sense. The only catch was, that the equations he used to describe the Metric Tensor involved a lot of calculus. I don't have a particularly strong background in calculus as I'm more of a linear algebra kind of guy. So please help explain to me what the Metric Tensor is but via linear algebra.
 A: Let's say you want to find the arclength of a curve in space. This means finding
$$ \int_a^b \|\gamma'(t)\| \,\mathrm{d}t $$
when the curve is parametrized by $\gamma:[a,b]\to\mathbb{R}^n$. Note that $\|\gamma'(t)\|^2$ is the dot product of the tangent vector $\gamma'(t)$ with itself. The dot product is an example of a ("nondegenerate") symmetric bilinear form which is also "positive-definite" (so $\mathbf{v}\cdot\mathbf{v}>0$ for nonzero $\mathbf{v}$). Every such positive-definite symmetric bilinear form is a dot product with respect to some basis, but in an arbitrary basis looks like $\mathbf{x}^TG\mathbf{y}$ for some symmetric matrix $G$.
Now let's say we have a 2D surface $\mathcal{S}$ in $\mathbf{R}^3$ parametrized by an open patch $U\subseteq\mathbb{R}^2$ via $f:U\to\mathcal{S}$. Then we can parametrize a curve in $\mathcal{S}$ using $U$. That is, let $\gamma:[a,b]\to U$ be a curve in $U$, then $f\circ\gamma$ is a curve on the surface $\mathcal{S}$. The arclength is then
$$ \int_a^b \|(f\circ\gamma)'(t)\|\,\mathrm{d}t=\int_a^b \|(Df_{\gamma(t)})\gamma'(t)\|\,\mathrm{d}t $$
The differential $Df_{\gamma(t)}$ is a $3\times 2$ matrix, when applied to the tangent vector $\gamma'(t)$ in the patch $U\subseteq\mathbb{R}^2$ yields the corresponding tangent vector $(f\circ\gamma)'(t)$ on the surface $\mathcal{S}\subset\mathbb{R}^3$.
We can rewrite $\|(Df_{\gamma(t)})\gamma'(t)\|$ as $\sqrt{v^TGv}$ where $v=\gamma'(t)$ is the tangent vector and $G=(Df_{\gamma(t)})^T(Df_{\gamma(t)})$ is a symmetric matrix (which depends on the point $\gamma(t)$ in the patch $U$). Indeed, $G$ can be interpreted as a matrix-valued function defined on $U$. It's purpose is to designate what the appropriate inner product is for tangent vectors $v$ at each point of $U$ to match that of the surface $\mathcal{S}$.
Note we can also find angles with inner products, as well as area, volume etc. by integrating. So the utility of $G$ is not limited to arclength; it really lets you compute anything "intrinsic" about $\mathcal{S}$ you want to (i.e. anything that doesn't depend on some arbitrary aspect of whatever parametrization you may use). Thus, instead of having the surface $\mathcal{S}$ and the parametrization $f:U\to S$, you can settle for having $U$ and the matrix-valued function $G$ on $U$ for all your calculational needs.
A: I would like to write an answer from the perspective of a physicist, since you mentioned Einstein Field Equations. Also, because could be a nice way to motive mathematical-oriented answers (like the one wrote by runway44 above). I will motivate the whole thing using the spacetime concept perspective.
I) Intuitive Idea
Intuitively and roughly speaking, spacetime is the "place" of all events, or the set of all events. An event is something that "happens in a time $\tau$ and takes place somewhere". You can grasp the main concept with a simple example: You have a physics test, Friday 11:00 AM, at the Physics Departament building, on floor 5. Well, if you go to the right place but wrong time you will miss the test. To access the event "test" you have to be in the right place at right time. So, you necessarily must to deal with four numbers: one for time and three for space.
Because of relativity, the time are not just a parameter, but a coordinate! In Lorentz transformations you transform time as a usual coordinate. You must consider time as just another coordinate as the usual spatial ones.
II) Dimension
The most elementary definition of dimension comes from a mathematical subject called linear algebra, which is one of the "mathematical tools" used to properly describe general relativity (GR) in mathematical terms. In GR, we basically deal with finite-dimensional vector spaces, so the concept of dimension is the most elementary one:

A dimension is the number of the basis vectors of a given vector space.

So a "mathematical 4th dimension" is just a 4-dimensional vector space.
III) Spacetime: A brief explanation
Well, here is where we use math to describe physics. The physics of spacetime was introduced in 1905 with Einstein's paper. But spacetime was born in 1906 with Minkowski's paper. Now, there are some facts that we will use to construct the proper idea of spacetime: 
1) In physics we can measure lengths and time and a mathematical object which have this property of "measure" is the norm given by a inner product.
In Newtonian mechanics, the norm is the Euclidian one:
$$ \|v\|^{2} := \langle v,v\rangle = \sum^{3}_{i=1}\sum^{3}_{j=1} \delta_{ij}v^{i}v^{j} \tag{1}$$ 
where $\delta_{ij}$ is the matrix:
$$ \delta_{ij} = 
\begin{bmatrix}
1&0&0\\
0&1&0\\
0&0&1\\
\end{bmatrix}
$$
In a sense, this norm together with a vector space gives the geometrical structure of Newtonian mechanics because we can calculate lengths, define vectors, calculate velocities and accelerations, and so on.... 
2) This norm sets what we call "Euclidean space" or "Euclidean geometry". Note that if you define another dimension, "the 4th dimension", you will just construct a 4-dimensional Euclidean space.
Now, the physical fact is: the geometry of spacetime isn't Euclidean, because we use a particular norm called the "Minkowski norm" or "Lorentz norm" on a 4-dimensional vector space. Because of this fact all the "conventional linear algebra" must to be adaptated to the Lorenztian geometry, given by the norm:
$$ \|v\|^{2} := \langle v,v\rangle = \sum^{3}_{\mu=0}\sum^{3}_{\nu=0} \eta_{\mu\nu}v^{\mu}v^{\nu} \tag{2}$$ 
where $\eta_{\mu\nu}$ is the matrix:
$$ \eta_{\mu\nu} = 
\begin{bmatrix}
-1&0&0&0\\
0&1&0&0\\
0&0&1&0\\
0&0&0&1\\
\end{bmatrix}
$$
III) Spacetime: The general picture
The matrix of inner product $(2)$ (and in general) is called the components of the metric tensor $g$. The metric tensor  is (roughly speaking) a bilinear map which produces a particular scalar called a line element, which is simply the value of the norm of differential line element vectors, i.e.

$$ ds^{2}\equiv g\Bigg(dx^{\mu}\frac{\partial \vec{r}}{\partial x^{\mu}},dx^{\nu}\frac{\partial \vec{r}}{\partial x^{\nu}}\Bigg) := \|d\vec{r}\|^{2} =: \langle d\vec{r},d\vec{r}\rangle = \sum^{3}_{\mu=0}\sum^{3}_{\nu=0} g_{\mu\nu}dx^{\mu}dx^{\nu} \tag{3}$$ 
Now, in general metric tensors aren't easy matrices like $\delta_{ij}$ and $\eta_{\mu\nu}$. In fact the metric tensor can become a tensor field which varies through space (and then the geometry varies pointwise).
To describe this general behaviour of "a tensor field which varies through space (and then the geometry varies pointwise)" we need the manifold mathematical framework (which is beyond the scope of this answer). 
$$ g_{\mu\nu} = 
\begin{bmatrix}
g_{00}(x^{\mu})&g_{01}(x^{\mu})&g_{02}(x^{\mu})&g_{03}(x^{\mu})\\
g_{10}(x^{\mu})&g_{11}(x^{\mu})&g_{12}(x^{\mu})&g_{13}(x^{\mu})\\
g_{20}(x^{\mu})&g_{21}(x^{\mu})&g_{22}(x^{\mu})&g_{23}(x^{\mu})\\
g_{30}(x^{\mu})&g_{31}(x^{\mu})&g_{32}(x^{\mu})&g_{33}(x^{\mu})\\
\end{bmatrix}
$$
With this manifold framework, we can give a sufficiently general description of spacetime:

A spacetime is a 4-dimensional manifold $\mathcal{M}$ with a pseudo-riemannian metric $g_{\mu \nu}$:
  $$ (\mathcal{M},g_{\mu \nu}) $$

IV) Spacetime: Merging the intuitive idea with math
So, spacetime is the stage of special relativity and general relativity. It tells you which events are in your future, in your past and the ones which you cannot access in a sufficiently small proper time (the time of the clock in your hand, the time of the observer at rest on his own reference system, in general a tetrad). Spacetime is also a geometrical 4-dimensional entity which tells you that because of the Lorentz signature you need to give spatial and temporal directions and, of course, the geometry isn't Euclidean anymore.
