What does "kernel" mean in integral kernel? In functional analysis, there is the term "integral kernel". Examples are Possion kernel, Dirichlet kernel etc. 
In algebra, the term kernel of a homomorphism refers to the inverse image of the zero element.
Are these two terms related? If not, where did the word "kernel" in the term "integral kernel" come from?
 A: I think it simply denotes the inner part.
According to dictionary, kernel is "the important, central part of anything".
(This is the third meaning in Chambers Concise Dictionary). From O.E. cyrnel=corn,grain + dimin. suffix -el).
I also know the kernel of an operating system.
A: An integral kernel is, of course, an integrable generalization $K(x,y)$ of a matrix $M_{j,k}$.  You could very loosely call this a "kernel" in the sense of the "core" of the formula for a integral linear operator.  For comparison, Wiktionary tells me that in German, a Kerngehäuse is an apple core, while Kernphysik is nuclear physics.
But mainly, I think that the two uses of kernel, one for the null space and one for an integration matrix, is just a terrible collision of terminology that became standard by accident.  It's nice when a mathematical term is an inspired metaphor or neologism.  For instance the word "spectrum" for the set of eigenvalues of an operator was not just inspired, but also prescient and profound.  (As I understand it, the term was chosen by mathematicians, by analogy with spectral lines in chemistry, shortly before the development of quantum mechanics.)  But sometimes we're just unlucky, or maybe collectively stupid.
As Jan Kolar points out, the kernel of an operating system is a third metaphorical use of the word that makes vastly more sense.
A: Both the kernel of a linear operator and the integral kernel come from the German word "Kern". Both are translations of it.
In German "Kern" means kernel, core, nucleus at the same time. Furthermore, for instance, the place where the seeds are in an apple are also called "Kern".
As in English, it refers to something central or essential (as in the integral or in the Earth (the core, "Erdkern")), but also as something hidden (as in an apple or for the linear operator).
It might be that this variety of meanings might be lost when you translate it into English.
A: http://jeff560.tripod.com/k.html

KERNEL. Ivar Fredholm used the French word, “noyau,” in his famous paper on INTEGRAL EQUATIONS, “Sur une classe des équations fonctionnelles,” Acta Math., 27, (1903), 365–390. David Hilbert put this into German as "Kern" in his “Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen,” Nachrichten von d. Königl. Ges. d. Wissensch. zu Göttingen (Math.-physik. Kl.) (1904) p. 49. The English word "kernel" appears in M. Bôcher’s Introduction to the Study of Integral Equations (1909): “K is called the kernel of these equations.” (quoted in the OED). See G. Birkhoff & E. Kreyszig (1984) “The Establishment of Functional Analysis,” Historia Mathematica, 11, 258-321.


Kernel was an established term in Fourier analysis by the time of A. Zygmund’s Trigonometrical Series (1935). A JSTOR search found the "Fejér kernel" and "Dirichlet kernel" in Charles N. Moore’s "On the Application of Borel’s Method to the Summation of Fourier’s Series" (Proceedings of the National Academy, 11, (1925), 284-287) but it is unlikely that this was the first published use of these terms.


Kernel entered Statistics with the use of Fourier theory in describing estimates for spectral density and probability density functions. A JSTOR search found E.  F. Schuster (Annals of Mathematical Statistics, 40, (1969), p. 1187) referring to "a so-called kernel class of estimates," introduced by M. Rosenblatt in "Remarks on Some Nonparametric Estimates of a Density Function," Annals of Mathematical Statistics, 27, (1956), 832-837. Earlier "Fejér kernel" was used in U. Grenander & M.Rosenblatt’s "Statistical Spectral Analysis of Time Series Arising from Stationary Stochastic Processes," Annals of Mathematical Statistics, 24, (1953), 537-558. An alternative term, especially popular in time series analysis, is WINDOW.

