Elliptical wing planform and elliptical lift distribution
Lift force over the wing is said to be of elliptical distribution if the lift per span ($L'$) along the wing span is of the following:

$$L'(y) = \rho_\infty V_\infty \Gamma_0 \sqrt{1-(\frac{2y}{b})^2}$$

where $\rho_\infty$ is density of the airflow, $V_\infty$ is the free-stream airspeed, $b$ is length of wing span, $y$ is lateral coordinate along the wing (0 being wing root), and $\Gamma_0$ is the circulation at wing root.

The above is essentially an equation of an ellipse, and hence the name elliptical lift distribution.

In low Mach flight, it can be theoretically demonstrated that an elliptical lift distribution produces the least induced drag for a given flat span. Induced drag of a lifting surface in incompressible flow can be expressed as (Ref Anderson, Fundamentals of Aerodynamics):

$$C_{D_i}=\frac{C_L}{\pi e A}$$

where $C_L$ is the lift coefficient, $A$ is aspect ratio and $e$ is the span efficiency factor.

For a flat span, $e$ must be smaller or equal to 1, and it's only equal to 1 when the lifting surface has an elliptical lift distribution.

When the wing has zero twist, an elliptical planform produces an elliptical lift distribution. However, elliptical lift distribution can also be coaxed by judiciously twisting the wing for a planform that is not geometrically elliptical.

Addendum on non-flat span:

In the case where the span is not flat, either because it has dihedral or is curved, it has been theoretically demonstrated that a non-flat span can achieve lower induced drag than a flat wing of equal projected span. This forms the theoretical basis behind winglets, end plates, spiroids, etc.

However, if we compare the non-flat wing to a flat wing whose length is that of the unfurled non-flat wing, the flat wing always has lower induced drag. That is, if there is no limit on the wing span length, elongating it is always better from an induced drag perspective.

Below is a graph comparing the induced drag of an optimal non-flat (circularly cambered wing) wing to that of an elliptically loaded flat wing. A value of 1 indicates no change. A value greater than 1 indicates more induced drag and a value less than 1 indicates less. $\beta$ is the span camber factor. 





Data and figure compiled from https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19630006412.pdf