I see orange, blue, purple and yellow on the page; what do they signify?
Magenta is unconstrained.
Blue is special case that can only exist for N=2.
The rest are yellow.
The number of lines in the middle show the number of touchs (constraints, 0 to 6).
Not sure how multiple solutions are shown (e.g. N=6 has two solutions).
It appears that circles are colored blue if their number of contacts (with either other circles or the outside) is equal to 1 or 2. They're magenta if their number of contacts is zero.
I'm not sure what the distinction between orange and yellow is-- it doesn't seem to be strictly number of contacts, because I can see cases where both orange and yellow have four contacts.
The colours are about constraints. Orange is a locked constraint between 3 circles (inner and outer included).
If you think about it, N=2 is the only solution that the circles are constrained to touch in two places (and N=1 is weird because it touches in infinite places).
At N=104, the blue circles are surely a drawing fault, they are actually unconstrained and should be magenta. Shown by fact that those blue circles have a dot in the middle (not touching), but a neighbour circle has a line pointing to the blue circle (touching), which is contradictory.
A magenta circle touches no other circles.
A blue circle touches one other circle.
An orange circle touches two other circles.
A yellow circle touches three other circles.
N the number of circles; colors correspond to active researchers in the past, see "References" at the bottom of the page
Unless specific circles in the diagram are somehow linked to the specific authors?
It probably isn't immediately applicable to every single person's life but it might help some industries squeeze another 0.1% out of their production line. The free work presented here might cover a couple engineer's salaries once they find this website.
OTOH, there are a lot of these math toys that start life in the abstract and then end up finding extremely practical applications down the line— pretty sure that was the case for a lot of the dusty corners of linear algebra until 3D graphics was suddenly a thing and it all became super relevant very quickly.
Also interesting that there are seemingly no solutions without loose circles beyond 91.
"The sequence of N's that establish density records" link leads to an empty page, but this sequence is also known as OEIS A084644 "Best packings of m>1 equal circles into a larger circle setting a new density record", and starts with 2, 3, 4, 7, 19, 37, 55, 85, 121, 147, 148, 150, 151, 187. https://oeis.org/A084644
Look for example at N=1759: http://hydra.nat.uni-magdeburg.de/packing/cci/cci1759.html
Compare it with N=1758, which has a slight "imperfection": http://hydra.nat.uni-magdeburg.de/packing/cci/cci1758.html
Or with N=1760, which is too "tight" resulting in a worse density: http://hydra.nat.uni-magdeburg.de/packing/cci/cci1760.html
But the references section seems to be missing entry  for some reason.
= 1/radius; an orange field means that David W. Cantrell's conjectured upper bound is violated
I'm sure we know the bound exists, but I'm more interested in whether we've found a closed form (or even just analytic) way to express the bound.
First glance doesn't show anything obvious, but I'm no mathematician.
I remember learning about N-dimensional sphere packing for coding.