In Evans' book on PDEs, his definition of an Elliptic Linear PDE is an equation of the form:
$$ Lu \equiv -\sum \limits_{i,j = 1}^{n} \left( a_{ij}(x) u_{x_i} \right)_{x_j} + \sum \limits_{i=1}^{n} b_i (x) u_{x_i} + c(x)u = f(x) $$
defined on an open bounded region $U$, where we assume for simplicity that $a_{ij} = a_{ji}$, and we have the condition that there exists a $\theta >0$ such that:
$$ \sum \limits_{i,j = 1}^{n} a_{ij}(x) \xi_i \xi_j \geq \theta |\xi| $$
for every $x$ in $U$ and every $\xi$ in $\mathbb{R}^n$.
What is the motivation for this definition?
One doesn't necessarily see what the definition is for when staring at it. The reason becomes apparent later, when the definition is used. So let's use it to prove that
$$-\sum \limits_{i,j = 1}^{n} \left( a_{ij}(x) u_{x_i} \right)_{x_j} = 0 \quad \text{and}\quad u_{|\partial \Omega}=0 \implies u\equiv 0$$ The above is an important property: if it fails, we don't have maximum principle and don't have uniqueness for the boundary value problems, i.e., don't have much of the theory of elliptic PDE.
Proof. Multiply the PDE by $u$ and integrate by parts (note: no boundary term because $u=0$ on the boundary): $$ 0 = -\int_\Omega \sum \limits_{i,j = 1}^{n} \left( a_{ij}(x) u_{x_i} \right)_{x_j} u = \int_\Omega \sum \limits_{i,j = 1}^{n} \left( a_{ij}(x) u_{x_i} \right) u_{x_j} $$ Now the ellipticity condition allows us to continue with
$$ \cdots \ge \int_\Omega \theta |\nabla u|^2 $$ and we conclude that $|\nabla u|\equiv 0$, hence $u$ is constant, and the constant must be zero because of the boundary condition.
Euler-Lagrange equation
If you are familiar with the calculus of variations, you can recognize that minimization of the functional $\sum a_{ij}(x) u_{x_i}u_{x_j} $ leads to the aforementioned PDE. In terms of this functional, the ellipticity condition is known as strong convexity, which is a standard assumption in optimization. It allows us to have some control over the behavior of the minimizing point, which simple convexity does not.