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Somewhere I've read the following:

A graph is a pair $(V, E)$ consisting of a set $V$ and a set $E\subseteq {V\choose 2}$ of edges.

What does the notation ${V\choose 2}$ mean? Where can I read a definition of this notion (specifically, is there a wikipedia page about it?)? I think it means the set of all $2$-element subsets of $V$.

Can one generally define ${A\choose \kappa}$, where $A$ is any set and $\kappa$ any cardinal, to be the set $\{T\subseteq A: |T| = \kappa\}$ (where $|T|$ denotes the cardinality of $T$)?

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    Yes I had a lecturer drop that piece of notation on me once too without any explanation and also found it disconcerting at first -- asking after the lecture, the explanation given was the same as yours. – 2017-02-22

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This was a new piece of notation to me too, but yes, I think it's safe to assume they mean "the set of all two-element subsets of $V$". And in general, yes, you could define it like that. Just as long as you explicitly define it before you use it so that you do not cause the same confusion in your reader as you have experienced here.

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    @PleaseHelp Note that I personally think it's a good notation, seing how binomial coefficients are related to all subsets of a given size already. For instance, you get lovely relations like $\left|\binom{V}{n}\right| = \binom{|V|}{n}$ (at least for finite $n$ and $V$; I haven't looked into the infinite ones). – 2017-02-22
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What does the notation V choose 2 mean?

Here it is given E is a subset of (V choose 2) . (V choose 2) means a complete graph with all edges(pair of vertices means there is a edge).All pair of vertices means all the edges in the graph .

Can one generally define (A k), where A is any set and κ any cardinal, to be the set {T⊆A:|T|=κ} (where |T| denotes the cardinality of T)?

The explanation is like this ..
You have a set of A elements and you choose k elements out of it . If you referring T as (A choose k) then you pick k elements out of A , so T is a subset of A and since T contains k elements its cardinality is k .