Good reference for studying operads? Can you, please, recommend a good text about algebraic operads?
I know the main one, namely, Loday, Vallette "Algebraic operads". But it is very big and there is no way you can read it fast. Also there are notes by Vallette "Algebra+Homotopy=Operad", but they don't have much information and are too combinatorial. So what I am looking for is a pretty concise introduction to the theory of algebraic operads, that will be more algebraic then combinatorial, and that will give enough information to actually start working with operads.
Thank you very much for your help!
Edit: I have also found this interesting paper Modules and Morita Theorem for Operads by Kapranov--Manin. Maybe it's a bit too concise for the first time reading about operads, but it has a lot of really nice examples and theorems.
There are also notes by Vatne (only in PostScript).
 A: Benoit Fresse's book Modules over Operads and Functors is masterful.
Additionally, here are a couple of very good survey articles and notes from conferences:
AMS "What is..." article written by Stasheff
Expository article by Shenghao Sun
Notes from Algebra, Topology, and Fjords Conference
A: The book of Markl, Stasheff and Shnider is also a standard reference. 
Also, a good jumping-in point could be Ginzburg and Kapranov's "Koszul duality for operads".
A: The book by M. Bremner and V. Dotsenko titled Algebraic Operads: an algorithmic companion (published in 2016) is (in my perhaps biased opinion) a must-have for those wishing to complement their reading of Loday--Vallette. As the authors explain :

It is fairly accurate to say that the aim of this book is to create an
accessible companion book to [180] which would, in the spirit of [64] contain enough hands-on methods for working with specific operads: making experiments, formulating conjectures and, hopefully, proving theorems, as well as, in the spirit of [252], include enough interesting examples to stimulate the reader toward those experiments, conjectures and theorems.

As the back-matter explains, it contains a systematic treatment of Groebner bases in several contexts, starting with non-commutative polynomials, and then moving to richer structure like twisted and shuffle algebras, and operads (ns, shuffle, symmetric), the main topic of the book. Like the book of Loday--Vallette, many instances of the book record relatively recent results concerning operads and related structures, and at the same time provides the reader with many challenging exercises (sometimes prompting them to use a CAS, if necessary) that provide invaluable insight for those aiming to make concrete computations using rewriting systems and their kin to study and prove results about operads.
[64] is Ideals, Varieties and Algorithms by Cox, Little and O'Shea,
[180] is Algebraic Operads by Loday and Vallette and
[252] is Combinatorial and Asymptotic Methods in Algebra by Ufnarovski.
A: Since both the following references appeared significantly later than the OP, it seems useful to add: 


*

*Donald Yau: Colored Operads. AMS. Graduate Studies in Mathematics
Volume 170

*The review of the above book written by Nick Gurski in the most recent issue (September 2017) of the Jahresbericht der deutschen Mathematiker-Vereinigung
A: A recent good book is Algebraic Operads by Jean-Louis Loday and Bruno Vallette.
