Intuition for Calderon-Zygmund operator? What is the best intuition for Calderon-Zygmund operators? Why are they so important in singular integrals, and complementary, which singular integrals don't they cover?
 A: The best intuition is the Hilbert transform which is defined for suitable class of functions $f$ as $\mathcal H f (x):= \lim _{\epsilon \to 0} \int_{|x-t|\geq \epsilon} \frac{f(t)}{x-t}~dt$. According to Marcel Riesz's famous convexity theorem, $\mathcal H$ is a bounded operator on $L^2(\mathbb R)$. The proof itself is of great importance, because it used complex method to prove a real theorem. The Hilbert transform is an important operator, for example if $\phi = u+iv$ is an analytic function, $\mathcal H$ relates the value of $u$ and $v$ on the line $y=0$. For more about the Hilbert transform see first chapter of M. Christ's book. S. Krantz says that (chapter 2 of the book cited):

The Hilbert transform is, without question, the most important
  operator in analysis.

With this brief introduction to the Hilbert transform, it is basic example of a (Calderon-Zygmund) singular integral operators, and the only one in the case $n=1$. Calderon and Zygmund generalized M. Riesz' idea to higher dimensions, and thus the theory of singular integral created. The immediate generalizations of Hilbert transform on higher dimensions called Riesz transforms: for $i=1,2,\dots n$ the operator $\mathcal R^i$ is defined as: 
$\mathcal R^i f (x):= \lim _{\epsilon \to 0} \int_{|x-t|\geq \epsilon} \frac{x_i -t_i}{|x-t|^{n+1}}f(t)~dt$. Riesz transforms $\mathcal R^i$ are also bounded on $L^2(\mathbb R^n)$.
You could find some historical background in a paper that E. Stein published in the Notices of AMS. There are also many applications of theory, for example in partial differential equations, geometric measure theory, and so on.
Stein, Elias M., Singular integrals: The roles of Calderón and Zygmund, Notices Am. Math. Soc. 45, No.9, 1130-1140 (1998). ZBL0973.01027.
Calderón, Alberto P.; Zygmund, Antoni, On the existence of certain singular integrals, Acta Math. 88, 85-139 (1952). ZBL0047.10201.
Krantz, Steven G., Explorations in harmonic analysis. With applications to complex function theory and the Heisenberg group, Applied and Numerical Harmonic Analysis. Basel: Birkhäuser (ISBN 978-0-8176-4668-4/hbk; 978-0-8176-4669-1/ebook). xiv, 360 p. (2009). ZBL1171.43001.

Christ, Michael, Lectures on singular integral operators. Expository lectures from the CBMS regional conference held at the University of Montana, Missoula, MT (USA) from August 28-September 1, 1989, Regional Conference Series in Mathematics. 77. Providence, RI: American Mathematical Society (AMS). ix, 132 p. (1990). ZBL0745.42008.. 
