In many different places, I could find the notion on ''(poly)logarithm sheaves''. As is indicated in the name of it, I guess that it should have something to do with (poly)logarithm function: $\mathrm{Li}_s(z)$. How are they related to each other? Any reference would be helpful, but it would be better if it requires less preliminaries.

$\begingroup$ Locally, the sheaf of logarithmic forms $\Omega^1(\log D)$ is generated by objects of the form $df/f$, where $f$ vanishes on $D$. In some open covering, there are local representations of this differential form as a logarithmic derivative, and this justifies the name. en.wikipedia.org/wiki/Logarithmic_form $\endgroup$ – Francesco Polizzi Jun 23 '16 at 20:43
I would suggest looking at Hain's article (arXiv version here)
R. Hain. Classical polylogarithms. Motives Proceedings, vol II, Proc. Symposia Pure Math 55.2, 1994.
Very roughly, one writes out a multivalued function on $\mathbb{P}^1\setminus\{0,1,\infty\}$ which takes values in ${\rm GL}_{n+1}(\mathbb{C})$ where the entries of the matrix are built from (poly)logarithms up to ${\rm Li}_n$. This is the fundamental solution of a linear differential equation. The monodromy of the differential equation gives rise to a local system on $\mathbb{P}^1\setminus\{0,1,\infty\}$. This is the $n$th polylogarithm local system (which can equivalently be regarded as a sheaf). Actually, doing this for all $n$ gives rise to a prolocal system which you might call the polylogarithm sheaf. Under the correspondence between local systems and representations of the fundamental group, this corresponds to the representation of $\pi_1(\mathbb{P}^1\setminus\{0,1,\infty\})$ on the completion of its group ring w.r.t. the augmentation ideal.
This has a Hodgetheoretic and even motivic refinement. These are relevant when trying to express regulators on Ktheory in terms of polylogarithms (on the way to formulas for special Lvalues).
You can find all this and more in Hain's article, also lots of references in there. Unfortunately, this is not quite the discussion that would "require less preliminaries"...

$\begingroup$ >Under the correspondence between local systems and representations of the fundamental group, this corresponds to... I don't think this is quite right  I think it's only a rather tiny quotient of the fundamental group. The whole prounipotent fundamental group corresponds to socalled "multiple polylogarithms," and the associated differential equation is usually called the "free KZ equation" if I understand these things correctly. $\endgroup$ – Daniel Litt Feb 9 '17 at 4:12