How should one call and use categories that are not locally small? In the paper I am now writing most of the categories are locally small (so, object form a class and $Mor(X,Y)$ is always a set). Yet they are not small, and sometimes I have to mention something like the category of additive functors from $C$ into abelian group. What can I say about the "size" of this functor category (to note that it is not locally small in contrast to other categories used in the paper)? What can one say about the functor category if $C$ is essentially small but not small? 
I appears that to treat categories of functors "properly" one has to use Grothendieck universes. Yet the papers (and a book) on triangulated categories that I cite in my text (almost) ignore this matter, and I don't want to pay much attention to it either. So I wonder which terminology can I use and which "precautions" should be taken so that my simple arguments concerning functor categories will be mathematically correct. In particular, if the restriction of certain functors to a small subcategory of $C$ gives an equivalence of functor categories then can I say that the original category of functors is locally small?
Do you know any text that introduces "the most conventional" terminology for these matters? I would not like to mention universes, and I don't like terms like "small set".
 A: I am not sure if this answers the question, but it was too long to be added as a comment.
For the terminology, you may see $\S.1.1.1$ in Kashiwara and Schapira's Categories and Sheaves. Such categories in question are called big categories, as $\mathrm{hom}$-sets are not small sets. Also, when $\mathcal{C}$ is not small, the catgeory $\mathop{\mathrm{Fun}}(\mathcal{C},\mathbf{Ab})$ is a very large category, meaning that its set of object is not even a class. Yet, as Marc Hoyois mentioned, $\mathop{\mathrm{Fun}}(\mathcal{C},\mathbf{Ab})$ is locally small when $\mathcal{C}$ is essentially small.
As you may be aware, the meaning of a category being (locally/essentially) small, (very) large, or big depends very much on the adopted foundations. And, the main issue with big categories is the lack of a $\mathrm{hom}$ bifunctor to $\mathbf{Set}$, and hence the Yoneda embedding and other notions depending on the $\mathrm{hom}$ bifunctor.
Concerning "I would not like to mention universes, and I don't like terms like "small set".", to be precise, one at least needs to specify what is meant by a set and proper class, and that in turn depends on the foundations.
Regarding triangulated categorises, the main set-theoretical issue I am aware of, is the question whether a Verdier quotient of a locally small category is locally small or not, and this is discussed, more generally for multiplicative systems, in proposition.7.3.3 in Categories and Sheaves.
A: As indicated by user337830, understanding your questions, hence their answers, depends on the foundational setting you use for mathematics.
Beyond a few remarks in MacLane's Category Theory for the Working Mathematician  (Chapter I, Section 6) and the initial setting of Kashiwara/Schapira's Categories and sheaves, the standard litterature is rather poor on this matter, possibly reflecting the absence of interest of “working mathematicians” about foundations. Grothendieck's SGA 4 gave slightly more comments in Expose 1 (2 sections on universes, and an Appendix by Bourbaki). More positively, this deficiency may reflect the fact that for “usual mathematicas”, the precise foundational setting is rather innocuous.
For example, Kashiwara and Schapira introduce universes, but forget them right away, as does Grothendieck. (In K-S's book, this leads to unprecise expressions, such as “for a sufficiently large universe”, and some change of universes in the middle of a proof I can't understand the meaning of...)
The most naive point of view consists of having all categories and functors viewed as formulas in the language of set theory. It is OK for a lot of things such as Grothendieck's Galois equivalence of categories (but not for categories of functors, I would guess). 
Then come universes or classes. The point of view of universes has the advantage of lying within classical (ZFC) set theory, while introducing classes requires another foundational system, Gödel-Bernays's (GB), say, something mathematicians may be reluctant to do.
As MacLane writes: if you have an universe $U$, you can call an element of $U$ a small set, and a subset of $U$ a class (or a large set), so that a model of ZFCU furnishes a model of GB.
For more complicated constructions, you'll need higher classes, or a (possibly infinite) sequence of nested universes. For example, categories of functors (categories of presheaves, for example) depend on the chosen universes; the fact that they are (or not) locally small is a theorem. I try never to forget that the fpqc-sheaf associated with a presheaf depends on the chosen universe (an example is given in Bosch/Lütkebohmert/Raynaud's book, Néron models).
Two papers sum up this with great clarity, I believe, and explain the merits/demerits of various approaches. One is due to Andreas Blass, The interaction between category theory and set theory, Contemp. Math. 30 (1984). I particularly like the (not so) recent preprint of Mike Shulman's, Set theory for category theory (2008, why is it still unpublished?).
In any case, I would advise you (if I ever may) to choose (possibly secretly) one foundational system you're OK with and to fix your terminology in consequence.
