See orange book by Adams.
A great introduction seems to be Chapter 13 of Handbook of AT, in Alg Top folder
http://ncatlab.org/nlab/show/loop+space
http://ncatlab.org/nlab/show/loop+space+object
http://ncatlab.org/nlab/show/connective+spectrum
http://ncatlab.org/nlab/show/infinite+loop+space
http://mathoverflow.net/questions/65103/uniqeness-of-loop-spaces
Jardine-Goerss chapter V.5 and V.6 describes the loop space of a simplicial group as left adjoint to a certain construction of a model for the classifying space.
Understand the infinite loop space machines of Thomason and May
Some good stuff might be in May’s books
Rings, modules, and algebras in infinite loop space theory, by Anthony D. Elmendorf and Michael A. Mandell http://www.math.uiuc.edu/K-theory/0748
Something on approximation by smooth manifolds: http://www.math.uiuc.edu/K-theory/0592
arXiv:1002.3636 Loop Spaces and Connections from arXiv Front: math.AT by David Ben-Zvi, David Nadler We examine the geometry of loop spaces in derived algebraic geometry and extend in several directions the well known connection between rotation of loops and the de Rham differential. Our main result, a categorification of the geometric description of cyclic homology, relates S^1-equivariant quasicoherent sheaves on the loop space of a smooth scheme or geometric stack X in characteristic zero with sheaves on X with flat connection, or equivalently D_X-modules. By deducing the Hodge filtration on de Rham modules from the formality of cochains on the circle, we are able to recover D_X-modules precisely rather than a periodic version. More generally, we consider the rotated Hopf fibration Omega S^3 –> Omega S^2 –> S^1, and relate Omega S^2-equivariant sheaves on the loop space with sheaves on X with arbitrary connection, with curvature given by their Omega S^3-equivariance.
May LNM0099 gives and exposition which seems old/early but maybe therefore easier to grasp.
LNM0161 Stasheff on H-spaces and loop spaces
LNM0196 contains historical survey on infinite loop spaces by Stasheff, really nice
Goerss-Jardine mentions that the loop space of a simplicial set should be the loops of a fibrant model, in other words the loop functor is a total right derived functor.
arXiv:1009.0804 The homotopy theory of function spaces: a survey from arXiv Front: math.AT by Samuel Bruce Smith We survey research on the homotopy theory of the space map(X, Y) consisting of all continuous functions between two topological spaces. We summarize progress on various classification problems for the homotopy types represented by the path-components of map(X, Y). We also discuss work on the homotopy theory of the monoid of self-equivalences aut(X) and of the free loop space LX. We consider these topics in both ordinary homotopy theory as well as after localization. In the latter case, we discuss algebraic models for the localization of function spaces and their applications.
nLab page on Loop space