's universal constant is approximately equal to 4.66920 16091 02990 67185 32038 20466 20161 72581 85577 47576 86327 45651 34300 41343 30211 31473 71386 89744 02394 80138...
The constant arises in situations where a function is repeatedly applied to its own output. Take for example the function which Feigenbaum first studied, f(x) = ax(1-x) on the domain 0<x<1. Say a is set to 2. Choose any x between 0 and 1 and calculate f(x), f(f(x)), f(f(f(x))), etc. Eventually this sequence will settle down to some fixed value. In fact you can easily tell what this value will be: since we set a to 2, you just need to solve x = 2x(1-x).
The Feigenbaum constant relates to how the behaviour of this function changes as we change the value of a. It turns out that when a increases past 3, the sequence of numbers x, f(x), f(f(x)), .... will tend to settle down to an alternating pattern; that is, it ends up oscillating between two values instead of getting stuck on one as was the case with a = 2. Increase a a bit further (to about 3.4495) and the sequence of iterated values will exhibit a "4-cycle", i.e., it ends up jumping between 4 different values instead of 2.
As a keeps increasing, the pattern continues: 8-cycles, 16-cycles, etc. emerge at certain values of a. Feigenbaum noted that the rate at which these cycles split in two gets increasingly faster: the splits (actually called bifurcations) occur at 3, 3.4495, 3.5441, 3.5644, and so on until chaotic behaviour emerges. Feigenbaum's number tells us how much faster the bifurcations occur each time: if you look at the gap between 3 and 3.4495, divide by the Feigenbaum number to get the gap between 3.4495 and 3.5441; divide by it again to get the gap between 3.5441 and 3.5644, and so on. Mathematicians will recognise this as the sum of a geometric progression; the limit point is the point at which chaos sets in.
Why all the fuss? Because this is a universal constant. If, instead of repeatedly iterating f(x) = ax(1-x) we used the function f(x) = a sin(x), we find that exactly the same constant describes the speedup rate of the bifurcations exhibited by this function.
For more information, search for "Feigenbaum" almost anywhere, or read Chaos by James Gleick for an exciting account of this discovery and others in the field of chaos theory.