Just so people know, the reason these are called elliptic is you can write the general form of a conic as

Ax^2 + Bxy + Cy^2 + Dx + Ey + F=0,

...for some constants A, B, C, D, E, and F, then an ellipse is where

B^2 - 4AC < 0.

Well, you can write the general form of a second order linear pde in two variables x and y as

Au_xx + Bu_xy + Cu_yy + Du_x + Eu_y+Fu = G.[1]

Where A, B, C, D, E, F, G are constants or functions of x and y. An elliptic PDE is where

B^2 - 4 AC < 0.

eg Laplace's equation (u_xx+u_yy=0) or the Schrodinger equation.

[1] In this notation, u(x,y) is the unknown function of x and y and u_xx denotes the second partial derivative of u with respect to x and you can extrapolate for the others.