Sources for Alexandrov surfaces There are two distinct notions in differential geometry associated
with A. D. Alexandrov: (1) Alexandrov spaces of courvature bounded
from below; (2) Alexandrov surfaces of bounded total curvature (more precisely integral of absolute value of Gaussian curvature is bounded).
(The confusion is reflected in our tag alexandrov-geometry.) Here
(1) was extensively studied in particular by Burago, Gromov, and
Perelman.
This question concerns the notion (2).  These surfaces were
extensively studied by Reshetnyak e.g.,
Reshetnyak, Yu. G.  On the conformal representation of Alexandrov
surfaces.  Papers on analysis, 287-304, Rep. Univ. Jyväskylä
Dep. Math. Stat., 83, Univ. Jyväskylä, Jyväskylä, 2001.
There is a nice survey article by Troyanov:
Marc Troyanov, Les surfaces à courbure intégrale bornée au sens
d'Alexandrov.  https://arxiv.org/abs/0906.3407
However, I haven't found anything like a complehensive or definitive
treatment of these surfaces, and am therefore looking for such a
reference.
 A: The most comprehensive is probably the survey of Reshetnyak (a yellow Encyclopedia book):
Reshetnyak, Two-dimensional Manifolds of Bounded Curvature, Encyclopaedia of Math. Sci., Vol. 70, Geometry IV, Springer, 1993.
Among people who worked on this recently are Thomas Richard and Clement Debin, see e.g. Clement's article A compactness theorem for surfaces with Bounded Integral Curvature.
A: I agree with the answer by Ivan Izmestiev, but let me add the original monograph
by Alexandrov and Zalgaller, 
Intrinsic geometry of surfaces, English translation,
American Mathematical Society, Providence, R.I. 1967. 
It is a different treatment of the subject in comparison with Reshetnyak, and it is also quite comprehensive.
Instead of potential theory of Reshetnyak, their main method is elementary geometry. Of the recent applications of this theory let me mention two
papers by M. Bonk:
MR2006006 
Bonk, Mario; Lang, Urs
Bi-Lipschitz parameterization of surfaces, 
Math. Ann. 327 (2003), no. 1, 135–169, and
MR1804531 Bonk, M.; Eremenko, A. Covering properties of meromorphic functions, negative curvature and spherical geometry. Ann. of Math. (2) 152 (2000), no. 2, 551–592.
