I read on Mathworld they are called half-closed interval; however, it doesn't tell me how to say it in English. Also, how does one denote which one of the two options that is the one the one refering too. (I don't know how to say that better).
$(a,b]$ is an interval from $a$ (exclusive - i.e. not including $a$ itself) to $b$ (inclusive)
$[a,b)$ is an interval from $a$ (inclusive) to $b$ (exclusive - i.e. not including $b$ itself)
For $(a,b]$ I say the interval from $a$ to $b$ including $b$ but not including $a$.
For $(a,b]$ I usually say "left-open interval a b". I'm not sure whether it is grammatically and semantically correct, but it is short and everyone understands.
The interval from $a$ to $b$, left-inclusive?
In general, I have no problem with a few more words for additional clarity. So 'the interval $a$, $b$, including the point $a$' seems fine to me too.
I would call $(a,b]$ the left open interval and $[a,b)$ the right open interval.
(Seems like I have the subconcious implicit assumtion that "normal intervals are closed"…)
$(a, b]$ is:
The interval $a$ to $b$ open on $a$ and closed on $b$
The interval $x$ such that $x$ is greater than $a$ and less than or equal to $b$.
Or any of the other answers given would be acceptable.
Although this is just a slight variation of the already posted pronunciations, I tend to say 'half-open interval from $a$ to $b$ without $a$' for $(a,b]$ and '(..)without $b$' for $[a,b)$