When was the continuum hypothesis born? The question Solutions to the Continuum Hypothesis states that the continuum hypothesis was posed by Cantor in 1890. In http://en.wikipedia.org/wiki/Continuum_hypothesis the year 1878 is quoted without source. What is the correct date? 
 A: Since your question includes the reference-request tag, I would point you towards the following two sources as well (also written by Moore):
Moore, G. H. (1988). The origins of forcing, Logic Colloquium ’86 (Frank R. Drake and John K. Truss,  editors). Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam,    143-173.
Moore, G. H. (1989). Towards a history of Cantor’s continuum problem. The history of modern   mathematics, 1, 79-121.
The latter is probably most hopeful for the question you have asked here, though I personally enjoyed the former, which I read through carefully when thinking about some of the issues outlined in this MO post. (I have listed two other sources on Cohen's related work and forcing there.)
Another nice source on CH and its "solution" is:
Yandell, B. (2002). The honors class: Hilbert's problems and their solvers.
I use scare quotes because the Continuum Hypothesis was ultimately shown to be independent from ZF(C), which means, in some sense, you need Z and F (and C) before you can talk about the Hypothesis in its resolved-form.
The Z part probably dates back to around 1908. See:
Zermelo, E. (1908). Untersuchungen über die Grundlagen der Mengenlehre. I. Mathematische  Annalen, 65(2), 261-281.
The F part probably dates back to around 1922.
For other dates and references, I think the Axiom of Choice book Asaf mentions is a nice source.
A: Finally, a good use for the newly purchased copy of "Zermelo's Axiom of Choice".
Moore writes that Cantor formulated the following problem in 1878:

Every infinite subset of $\Bbb R$ is either denumerable or has the power of the continuum.

The given reference is:

Cantor G. "Ein Beitrag zur Mannigfaltigkeitslehre." Journal für die reine und angewandte Mathematik 84 (1878), pp. 242-258.

Although searching for the reference, it seems that it may have published in 1877, so it's unclear which one is the correct date. The paper was written in 1877, though. Moore points that the hypothesis was given on page 257, but I don't read German very well (or at all, for that matter), so I can't tell.
Moore also adds that in 1895 Cantor pointed out that in $\aleph$ notation, which introduced in the paper below, that this is equal to $2^{\aleph_0}=\aleph_1$. Cantor already made this observation in 1882 in a letter to Dedekind, where he used the terminology "numbers of the second number-class" to talk about $\aleph_1$.

Cantor G. "Beiträge zur Begründung der transfiniten Mengenlehre." Mathematische Annalen 46 pp. 481-512.

However it is good to note that this equivalence requires the axiom of choice. The question you cite refers to this statement, and not the first statement.
(Both the statements appear on page 41 of the book "Zermelo's Axiom of Choice: Its Origins, Developments & Influence" by Gregory H. Moore)
A: The main theorem of Cantor's "Beitrag" (contribution) is a bijection between $\mathbb R$ and $\mathbb R^n$, as well as a bijection between $\mathbb R$ and $\mathbb R^{\aleph_0}$.   He uses (in modern language) Hilbert's Hotel to find a bijection between the real numbers and the irrational numbers (in the unit interval), and then uses continued fractions to find a bijection to Baire space $\mathbb N^{\mathbb N}$.  The bijection between Baire space and its square is then easy. 
At the end of the paper he introduces what we now call the continuum hypothesis: 
Cantor's original text, dated July 11, 1877: (from the link quoted by Asaf)
[...] Verstehen wir unter einer linearen Mannigfaltigkeit jeden denkbaren Inbegriff unendlich vieler, voneinander verschiedener reeller Zahlen, fragt es sich, in wie viel und in welche Klassen die linearen
Mannigfaltigkeiten zerfallen, wenn Mannigfaltigkeiten von gleicher Mächtigkeit in eine und
dieselbe Klasse, Mannigfaltigkeiten von verschiedener Mächtigkeit in
verschiedene Klassen gebracht werden. [...]
Durch ein Inductionsverfahren, auf dessen Darstellung wir hier nicht näher eingehen, wird der Satz nahe gebracht, dass die Anzahl der nach diesem Eintheilungsprinzip sich ergebenden Klassen eine endliche, und zwar, dass sie gleich  zwei ist. [...]  
Eine genaue Untersuchung dieser  Frage verschieben wir auf eine spätere Gelegenheit.
My translation
...If we define a linear manifold to be any embodiment of infinitely many different real numbers, the question appears, in how many (and which) classes the linear manifolds can be divided, if manifolds of the same power are placed in the same class, manifolds of different power in different classes. [...]
Using an inductive method, the presentation of which we will not address here, the theorem suggests itself that the number of classes resulting from this division is finite, and in fact two. [namely, the countable sets and those which can be bijected onto the unit interval]
We defer a closer investigation of this question to a later occasion. 
