Looking at lenses, specifically sigma macro lenses, I noticed that it had a ""Flat Field" front lens element". What does this mean? And how is it different to the norm?
The norm is a curved field.
When a lens is focused on a flat subject, the light rays from the subject will converge some distance behind the lens, at the focal point. Rays from different points on the subject - top, bottom, left, right - will converge at different points on the other side of the lens, these focal points together make up the focal plane.
The point is that the focal plane for a normal lens is curved, like in this illustration from wikipedia:
The vertical black line to the right represents a flat sensor, the arc is the focus plane. Source.
Combining a curved focus plane and a flat sensor, the net effect is that you can't get the edges and the center in focus at the same time.
A "flat field" lens tries to compensate so that the focal plane becomes flat rather than the normal fishbowl shape.
For normal lenses, subjects and working distances the curved field may not matter much, since we have enough DOF to cover the difference.
But at macro distances the DOF is very shallow, so the difference becomes visible. Compounding the problem, macro lenses were historically used in large part for reproduction of flat subjects like stamps or documents, so having the whole image in focus was critical.
And that's why you will mostly see flat field focus on macro lenses, because it is primarily at macro distances that the benefits start to outweigh the costs.
If you have a flat piece of paper with gridlines on it, position it so that it is parallel with the sensor (perpendicular to the lens), and focus at the center of the image, most lenses will show "soft" corners. The degree to which they are soft, or sharp, is a measure of quality, but most will be soft to some degree.
Macro lenses tend to be prime and because the focal length is fixed it gives manufacturers the ability to create a "flat" field. It's probably not perfectly flat, but merely perceptively flat. There may be some other reason, specifically related to the macro-ness of a lens, for why macro-primes are flatter than other primes, but I wouldn't know about that. The article linked by John Cavan may have an answer to that:
I understand the first part (more prominent at low magnification), but the explanation for it is beyond me.