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The axiom of countable choice ($CC$), also called $AC_\omega$ or $AC_N$, is a weak form of the axiom of choice; it says that the set of natural numbers is a projective object in Set. (Recall that the full axiom of choice states that every set is projective.)
In classical mathematics, countable choice is usually accepted because the full axiom of choice is accepted. In constructive mathematics the situation is more subtle. For varying reasons, some schools of constructive mathematics accept countable choice (though they reject the full axiom of choice). On the other hand, countable choice is not valid in the internal logic of a general topos, so if one desires this level of generality then it should not be assumed. There are also philosophical constructivist arguments against it. Fred Richman (RichmanFTA) has said that
Countable choice is a blind spot for constructive mathematicians in much the same way as excluded middle is for classical mathematicians.
All the reasoning in this page is constructive.
More explicitly, let $X$ be any set and let $p\colon X \to \mathbf{N}$ be a surjection. Then the axiom of countable choice states that $p$ has a section. If you phrase the axiom of choice in terms of entire relations, then countable choice states that any entire relation from $\mathbf{N}$ to any other set contains (in the 2-poset Rel) a functional entire relation.
Here we collect some consequences of the countable axiom of choice.
The Cauchy real numbers and Dedekind real numbers coincide (the law of excluded middle also separately implies this).
The Rosolini dominance is a dominance (the law of excluded middle also separately implies this).
Unlike the full axiom of choice, countable choice is often considered to be a constructively acceptable principle. In particular, it does not imply the principle of excluded middle. It is a consequence of COSHEP. A stronger version of countable choice, also a consequence of $COSHEP$, is the axiom of dependent choice ($DC$). In general, $DC$ is enough to justify results in analysis involving sequences.
Sometimes in foundations it is useful to consider a weaker version of countable choice, called $AC_{00}$. This states that any entire relation from $\mathbf{N}$ to itself contains a functional entire relation. In terms of surjections, this states that any surjection $p\colon X \to \mathbf{N}$ has a section if $X$ is a subset of $\mathbf{N} \times \mathbf{N}$ and $p$ is the restriction to $X$ of a product projection. $AC_{00}$ is enough to prove that every Dedekind real number is a Cauchy real number (the converse is always true).
The axiom of weak countable choice ($WCC$) states that a surjection $p\colon X \to \mathbf{N}$ has a section if, whenever $m \neq n$, at least one of the preimages $p^*(m)$ and $p^*(n)$ is a singleton. $WCC$ follows (for different reasons) from either $CC$ or excluded middle. On the other hand, $WCC$ is enough to prove that the fundamental theorem of algebra in the sense that every non-constant complex polynomial has a root; see Bridges et al (1998).
An even weaker form of countable choice was proposed by Martin Escardo; it states that any surjection of the form $A \sqcup (\mathbf{N}\times B) \to \mathbf{N}$ has a section, where $A\to \mathbf{N}$ is a decidable subset and $B$ is an arbitrary set with $\mathbf{N}\times B \to \mathbf{N}$ the projection. This follows from WCC and also from the limited principle of omniscience; see the constructivenews discussion.
Douglas Bridges, Fred Richman, and Peter Schuster (1998). A weak countable choice principle. PDF AMS PDF
Martin Escardo et. al., Special case of countable choice, message and discussion to the constructivenews list, google groups
Fred Richman, The fundamental theorem of algebra: a constructive development without choice. pdf