Who introduced the terms "equivalence relation" and "equivalence class"? Consider that the question does not concern the origin of the ideas of equivalence relation and equivalence class. It exactly concerns the origin of the terms "equivalence relation" and "equivalence class". It seems that the terms weren't in use at least until 1903 where Russell writes:

Peano has defined a process which he calls definition by abstraction,
  of which, as he shows, frequent use is made in Mathematics. This
  process is as follows: when there is any relation which is transitive,
  symmetrical and (within its field) reflexive, then, if this relation
  holds between u and v, we define a new entity Ø (u), which is to be
  identical with Ø (v).
Relations which possess these properties are an important kind, and
  it is worthwhile to note that similarity is one of this kind of
  relations.

UPDATE: Thanks to the suggestions given in the comments and a very informative answer of Francois Ziegler, suddenly the following piece from Russell's Introduction to Mathematical Philosophy  came into a new light: 

The question “what is a number?” is one which has been often asked,
  but has only been correctly answered in our own time. The answer was
  given by Frege in 1884, in his Grundlagen der Arithmetik. Although his
  book is quite short, not difficult, and of the very highest
  importance, it attracted almost no attention, and the definition of
  number which it contains remained practically unknown until it was
  rediscovered by the present author in 1901.

So, perhaps that was Frege himself who used the terms! Although, I couldn't find anything in Frege's writings yet. But now, considering Eugen Netto's paper (see Francois' update below) a very more important question now is: Is he indeed Russell who should be credited with rediscovering Frege in 1901?!  
PS. I am well aware that asking a new question inside another question is not a good idea. However, up until this post I had the feeling that apart from the origin of the terms I know "everything" about the history of the notions of equivalence relation and equivalence class (part of which has been published here). But, this new information caught me by surprise, and I could not help myself to add the new question. You may just ignore it.    
 A: Von Neumann uses "equivalence class" in Zur Prüferschen Theorie der idealen Zahlen, Acta Sci. Math. (Szeged) 2 (1926) 193-227, p. 197 (viewable after free registration):

Wir nennen $R$ und $S$ äquivalent, in Zeichen: $R\sim S$, wenn (...)
Satz 2. Es ist stets $R\sim R$. Aus $R\sim S$ folgt $S\sim R$. Aus $R\sim S$ und $S\sim T$ folgt $R\sim T$.
(...)
Infolge des Satzes 2. zerfällt die Menge der Folgen realer Zahlen in paarweise elementefremde Klassen untereinander äquivalenter Folgen.
(...)
Definition 4. Eine Aequivalenzklasse, die lauter Fundamentalfolgen enthält, nennen wir eine ideale Zahl.

This seems like an early example, in that he sees fit to add here the footnote: "We define the ideal number as the corresponding set of fundamental sequences, itself; naturally one could also regard it as an ideal element attached to this set."
I note also that Hasse's book Höhere Algebra (where Henry Cohn found the earliest occurrence of "equivalence relation" so far) appears to have a 1926 edition too.

Update 1: One might also quote a paper by Eugen Netto, Über die arithmetisch-algebraischen Tendenzen Leopold Kronecker's, in: Mathematical Papers read at the International Mathematical Congress (Chicago, 1893), Macmillan 1896, pp. 243-252, who writes:

"Jede wissenschaftliche Forschung geht darauf aus, Aequivalenzen festzustellen und deren Invarianten zu ermitteln (...)."
Jede Abstraction, z. B. die von gewissen Verschiedenheiten, welche eine Anzahl von Objecten darbietet, statuirt eine Aequivalenz; alle Objecte, die einander bis auf jene Verschiedenheiten gleichen, gehören zu einer Aequivalenzclasse, sind unter einander aequivalent, und der aus der Abstraction hervorgehende Begriff bildet die "Invariante der Aequivalenz."


Update 2: Finally(?) one should probably also quote Vol. 2 of Weber's Lehrbuch der Algebra (1896). Literally he uses neither "equivalence relation" nor "equivalence class", but see how close he gets in §152 "Aequivalenz":


*

*Zwei ganze oder gebrochene Functionale $\varphi$, $\psi$ im Körper $\Omega$ heissen äquivalent, wenn (...)


*Zwei Functionale, die mit einem dritten äquivalent sind, sind auch unter einander äquivalent.
Theilt man hiernach alle Functionale des Körpers $\Omega$ in Classen ein, indem man zwei Functionale in dieselbe oder in verschiedene Classen wirft, je nachdem sie äquivalent sind oder nicht, so ergiebt sich (...)

Same for Dedekind's Ueber die Theorie der algebraischen Zahlen (1879), §175:

Wir wollen nun zwei Ideale $\mathfrak a$, $\mathfrak a'$ äquivalent nennen, wenn (...)
Zugleich ergiebt sich hieraus, dass (...) je zwei Ideale $\mathfrak a'$, $\mathfrak a''$, die mit einem dritten Ideal $\mathfrak a$ äquivalent sind, stets auch miteinander äquivalent sein müssen. Auf diesem Satze beruht die Möglichkeit, alle Ideale in Idealclassen einzutheilen; (...) der Inbegriff $A$ aller mit $\mathfrak a$ äquivalenten Ideale $\mathfrak a$, $\mathfrak a'$, $\mathfrak a''$ (...) nennen wir eine Idealclasse oder kürzer eine Classe

(Of course, calling class a family of objects related by some equivalence is a custom that can be traced to much older work — cf. Dirichlet [1863, p. 172] or Eisenstein [1847, p. 118] or Gauss [1801, §223].)
A: Re: the two questions in your UPDATE, I'd like to recommend the 2005 paper Frege's Natural Numbers: Motivations and Modifications by Erich Reck:


*

*Regarding other people's awareness of Frege's definition of number, Reck involves — as if by coincidence! — two of our previous protagonists, p. 276:  

the Frege-Russell conception of numbers (...) seems to have been in the air already before Russell's writings. For example, the mathematician Heinrich Weber proposes essentially the same conception, independently of both Frege and Russell, in an 1888 letter to Richard Dedekind (...)


*Regarding Frege's awareness of "equivalence relations", he raises (but leaves open) precisely that question in endnotes 5 and 41:

It would be interesting to know to what degree, and in what form exactly, Frege was also aware of equivalence class constructions in the algebra of his time (...) I am not aware of any historical account of the use of such methods in nineteenth-century mathematics, or even earlier; thus I am not sure how safe it is, in the end, to assume that Frege knew about them.

A: This is to add something to Francois' answer to the UPDATE. This is from "Frege : philosophy of mathematics" by Dummett (1991), p. 50:

One of the mental operations most frequently credited with creative
  powers was that of abstracting from particular features of some object
  or system of objects, that is, ceasing to take any account of them. It
  was virtually an orthodoxy, subscribed to by many philosophers and
  mathematicians…, that the mind could, by this means, create an object
  or system of objects lacking the features abstracted from, but not
  possessing any others in their place. It was to this operation that
  Dedekind appealed in order to explain what the natural numbers are.
  …Frege devoted a lengthy section of Grundlagen, §§29-44, to a detailed
  and conclusive critique of this misbegotten theory; it was a bitter
  disappointment to him that it had not the slightest effect.

I had never a chance to read Frege's Grundlagen. Worse, at the moment, I don't know how  Reck, Dummett and Russell views about Grundlagen may come together! 
