What made the recent supermoon so super? I understand that a "regular supermoon" is a full moon when the moon is at perigee (closest to earth in its orbit). 

They seem to happen often, here's a post from one last year. But I've seen news articles saying this is the 'superest' super moon since 1948 and we won't see another super moon this bright until 2034.

What makes the recent super moon so bright?
The thing you have to really understand is that not all supermoons are equal. The official definition of a supermoon, as you state, is that the full moon must coincide with the moon being at perigee.

However, this never actually happens! The moon being at perigee happens for only a single instance of time - less than a second. The moon being officially "full", again only happens for a single instance of time (i.e., when the Sun, Earth, and Moon all form a single line). The chances of these two instances of time happening simultaneously are almost non-existent.

So, when someone calls a full moon a supermoon, they really just mean that the exact full moon instance is "pretty close" to the exact moment of reaching perigee. The closer the better and the more super your supermoon will be.

If you take a look at the Supermoon which occured on Nov. 14, widely proclaimed to be the "superest" supermoon for a long time, you can see that perigee was reached precisely at 11:23 UTC, while the full moon occurred precisely at 13:52 UTC, two and a half hours later. For pretty much every other supermoon, these two events are close enough for the moon to be called a supermoon, but not two-and-a-half-hours close. The closeness in time between reaching perigee and being truly full made this moon especially big and more important, especially bright.

There is also another thing which made this supermoon the superest and that was the actual distance of the Moon's perigee. As with anything in our solar system, the Moon's orbit is not perfectly elliptical, but instead has small variations and perturbations over time. This means that the perigee of the Moon's orbit is slightly changing over time. The November supermoon occured during a particularly close perigee that won't be matched until 2034 supermoon which you reference.