Irradiance Map Equation What is "dω"?
$E_n$ is illuminance (or irradiance), $L_i$ is luminance (or radiance) in direction $\omega_i$ and $d\omega$ is differential solid angle.

TL;DR: $d\omega_i$ is the area of a pixel in the cubemap projected onto the unit sphere.

And then a longer explanation (:

Solid angle is essentially the projected area of an object on a unit sphere and has unit of steradians (sr). Entire unit sphere has surface area of $4\pi$ and hemisphere has the area of $2\pi$, which comes up a lot in lighting calculations due to integrals over spheres and hemispheres respectively.

If you would want to calculate solid angle for an object and point $p$, you would project the object on the unit sphere at $p$ and calculate the area of the projection. Further the object is from $p$ smaller the solid angle gets. As an example, we can calculate solid angle for Sun as seen from the Earth. We know that Sun has about 0.52 degrees subtended angle from Earth, so we can calculate that the solid angle is:
$$2\pi*(1-Cos(0.52/2))\simeq0.00006469sr$$

So the differential solid angle is just an infinitesimally small solid angle. If you would numerically integrate over a sphere ($4\pi$ steradians) by taking $N$ uniform samples on the sphere, your $d\omega$ would be $4\pi/N$. With integrals $N$ goes to infinity, while with numerical approximation of the integral $N$ is some "sufficiently large" number.

However, in this paper you refer to they perform integration by rendering to a cubemap at $p$ and then iterating through all the rendered pixels. But because these pixels are rendered onto planes (cubemap faces), their projection on the unit sphere isn't constant, so $d\omega$ needs to be weighted by the position in the rendered image, which they explain in your second quote:


  Computing the solid angle dωi subtended by a particular texel is a
  little more involved. Texels near the center of the texture subtend a
  somewhat larger angle, while texels near the corner take up a smaller
  solid angle when we map the cube texture onto a sphere.