How is the notation $X\sim N(\mu,\sigma^2)$ read? Is it $X$ follows a normal distribution? Or $X$ is a normal distribution? Or perhaps $X$ is approximately normal..
What if there are several variables that follow (or whatever the words is) the same distribution? How is it written?
The variable X is distributed according to the Normal distribution with mean vector $\mu$ and standard deviation $\sigma$.
As regards the use of symbols $\sim$ ("follows", "is distributed according to "), and $\approx$ ("equals approximately"), see this answer. This is how the symbols are used at least in Statistics/Econometrics.
As regards the notational conventions for a distribution, the normal is a borderline case: we usually write the defining parameters of a distribution alongside its symbol, the parameters that will permit one to write correctly its Cumulative distribution function and its probability density/mass function. We do not note down the moments, which usually are a function of, but not equal to, these parameters.
So for a Uniform that ranges in $[a,b]$ we write $U(a,b)$. The mean of the distribution is $(a+b)/2$ while the variance is $(b-a)^2/12$. For a Gamma (shape-scale parametrization), we write $G(k,\theta)$. The mean is $k\theta$ and the variance $k\theta^2$. Etc.
In the case of the normal distribution, the parameter $\mu$ happens to also be the mean of the distribution, while the parameter $\sigma$ happens to be the square root of the variance. It is my (possibly mistaken) impression that in Engineering circles one sees more often $N(\mu, \sigma)$ (which conforms with the general notational rule), while in Econometrics circles almost always one sees $N(\mu, \sigma^2)$ (which falls to the temptation of providing the moments, by treating $\sigma^2$ as the base parameter and not as the square of it).
EDIT: My previous answer failed to answer the actual question. What follows is my attempt at a more to the point response.
How is the notation $X \sim N(\mu,\sigma^2)$ read?
Other answers already tell you what the notation means, namely that $X$ is a normally distributed random variable with some mean $\mu$ and variance $\sigma^2$. Dilip's answer also gives a nice account of what other possible interpretations there are when the notation is less clear than $\sigma^2$, e.g. for general parameters $\{a,b\}$, viz. $X\sim N(a,b)$.
Whenever I see this notation in text I tend to read it so that it makes sense grammatically. I would claim that this the sensible way to treat the notation. Thus, the answer to your question is that, knowing what the notation means mathematically, you simply read it in any way that fits the text. Here are a two examples:
(1) Let $X \sim N(a,b)$...
(2) Consider three independent random variables, $X\sim N(0,1), Y\sim N(1,2), Z \sim Exp(\lambda).$
In (1) I read it as (e.g.) "Let $X$ be normally distributed with mean a and variance b...", and in (2) I read it as "... $X$ is standard normal...".
Is it X follows a normal distribution?
Yes that works, too. Many people say it this way, although you might want to include the mean and variance characterizing the distribution.
Or X is a normal distribution?
No, that is incorrect. See this old answer of mine for an account of what a distribution is.
Or perhaps X is approximately normal..
No, that is also incorrect. There are other ways to denote this. As pointed out in the comments, $\overset{\cdot}{\sim}$ is one of them.
What if there are several variables that follow (or whatever the words is) the same distribution? How is it written?
If they are all independent, one easy way to write this is $X_i \overset{iid}{\sim} N(\mu,\sigma^2),i=1,2,\dots n$, given that you have $n$ variables (iid stands for independent and identically distributed). If they are not independent, you can say that $X_i, i=1,2,\dots,n$ are possibly dependent, but (marginally) identically distributed as $N(\mu,\sigma^2)$. Or you may have to instead declare their joint distribution -- that depends on what purpose you have for considering the random variables.
If they are jointly normal, it's easy to write that $\mathbf X :=(X_1,\dots,X_n)'\sim N(\mu, \Sigma)$ to fully characterize their joint distribution using some mean vector $\mu$ and covariance matrix $\Sigma$.
In general, you may define any multivariate distribution function $F$ and then write that $\mathbf X \sim F$.
The difficulty is not in knowing what $\mathcal N(\mu,\sigma^2)$ means. Even $\mathcal N(3,5^2)$ is reasonably unambiguous to most peaople as meaning a normal random variable with mean $3$ and variance $5^2$ or variance $25$ (purists should believe that the standard deviation is a more fundamental parameter than the variance should free to say "standard deviation $5$" instead). However, what is meant by $\mathcal N(a,b)$, e.g. $\mathcal N(3,25)$ is subject to at least three different conventions with respect to the variance or standard deviation. All three conventions agree that the $\mathbf 3$ is the mean $\mu_X$ of $X$ but the $\mathbf 25$ has different meanings to different people.
$X \sim \mathcal N(\star,25)$ means that the standard deviation of $X$ is $25$.
$X \sim \mathcal N(\star,25)$ means that the variance of $X$ is $25$.
$X \sim \mathcal N(\star,25)$ means that the variance of $X$ is $\dfrac{1}{25}$.
See this question and the comments that follow for some details.
$X$ is a random variable "$X$";
$\sim$ is read "is distributed as";
$N$ is read "Normal";
$\mu$ is read "with mean $\mu$" (the convention is that the first entry after the open parenthesis is the mean, and the second is the variance or standard deviation, depending on notation -- see below); and
$\sigma^2$ is read "with variance $\sigma^2$ (or standard deviation $\sigma^2$, depending on the usage of the author/user. In this case, I'm guessing it's with variance $\sigma^2$.
Putting it all together, you have a random variable $X$ which is distributed as Normal with a mean "mu" ($\mu$) and variance "sigma squared" ($\sigma^2$).
You can also say $X$ follows a normal. . .
If several variables follow the same distribution, you can represent this several ways, but you might want to index the variables from $i=1$ to $n$. Then you could write, $X_i\sim N(\mu, \sigma^2)$, for $i=1$ to $n$.
$X$ is Normally distributed with mean $\mu$ and standard deviation $\sigma$. The tilde does not mean approximation, as it is not related to an equals sign, though implies it in a way since X is never definitively known.
