That is, it's twice the radius where the radius is from the centre of the sun to some edge. But what is that edge?
Fusion reactions taking place inside the core of the star produce a huge amount of energy, most of which becomes heat. These reactions are not evenly distributed through the star and so there are phenomena such as sun spots and solar flares, however the total amount of energy produced tends to be reasonably constant.
I would say that the edge is defined by the average point where the gravity reaches equilibrium with the pressure of the star's super-heated gases (as a result of internal fusion).
See the picture of the Sun on Wikipedia
That edge/balance will change when the sun begins to run low on hydrogen. At this time, the reactions inside the star will change causing it to become become a giant red star.
I guess you could compare it with the surface of sea water on Earth. It's technically not still and stable, but we can calculate an average value of the sea level. And it is because it's an average value that we can rely on that to determine altitude and earth radius as well.
Most literature will define the diameter of the Sun up to the photosphere, the layer of the solar atmosphere you would see if you were to observe the Sun in white light.
Of course the true edge of the solar atmosphere could be considered as the heliopause, where the direct influence of the Sun's magnetic field and solar wind end and interstellar space begins.
I thought I'd contribute an answer because there's a very recent paper on the subject:
It appeared in my RSS feeds this morning! A related writeup is online at the HMI website.
To answer the question, this measurement uses the transit of Venus to fit the limb-darkening law of the Sun. That is, the Sun is a bit fainter the further from the centre that you look. As you reach the optically thinner layers near the "surface", the brightness falls off rapidly, towards zero in the vacuum of space. The inflection point of the curve (as a function of distance from the centre of the disk) is a reasonable estimate of the "radius". As pointed out elsewhere, the value changes depending on which wavelength you use, but only by a few hundred km, compared to the Sun's overall radius of about 700 000km (actually more like 695 946 km), so the uncertainty is at or below the 0.1% level. Phil Plait wrote about a similar measurement (by the same team, I believe) that used the transits of Mercury in 2003 and 2006.
Finally, the team also used the limb-darkening (I think) to measure how round the Sun is. i.e. the diameter from top-to-bottom versus left-to-right. Answer: the Sun is very very round, with the radii differing by a few parts per million.
Look at the Sun. You shouldn't do this directly with the naked eye, but you can do so through a very dark filter, or project a suitably dark image through a pinhole. You can even find photos of the Sun on the internet.
What you see is a disk, uniformly bright and with a sharp boundary, surrounded by a comparatively much darker sky. The bright region is the part we consider the Sun, and that's how we get the radius.