Which standard deviation should I use in Student's t-test? There is something I'm not understanding about the Student's $t$-test, so I will write here an example to explain my doubt. I will write some considerations about the several steps: feel free to comment them if you think I'm wrong, also in details.
Let's say I know that the mean absolute B-magnitude of the population of elliptical galaxies is $\mu_{pop} = -21.3$ (absolute magnitudes are often negative numbers, and the more negative is the more luminous a galaxy is, but it doesn't really matter here). This value could come from decades of observations of thousand of galaxies or from tested theories (isn't it the same?).
Now I observe a sample of $N_{samp} = 13$ elliptical galaxies, finding a mean magnitude $\bar{x}_{samp} = -20$ with standard deviation $s_{samp} = 0.5$.
So I want to conduct a $t$-test in order to know if this sample is taken from the population. The null hypothesis here is that there is no relation between the two (i.e between the cosmic population of elliptical galaxies and from my observed sample).
The main doubt is: which standard deviation have I to put as denominator in $t$ definition? Given that
$$ t = \frac{| \bar{x}_{samp} - \mu_{pop} |}{\sigma} \, \mbox{,}$$
which $\sigma$ should I use? If I don't know the original (or we can say true?) standard deviation of the population $\sigma_{pop}$, I've readen that I can assume that $\sigma_{pop} \simeq s_{samp}/\sqrt{N_{samp}}$. But isn't true that, for $N_{samp} \to \infty$, I should expect $\bar{x}_{samp} \to \mu_{pop}$ and $s_{samp} \to \sigma_{pop}$? For what I've written the paradox is that $s_{samp} \to \sigma_{pop}$ for $N_{samp} \to 1$.
And if have also, as is in the case of my example, the value of $\sigma_{pop} = 1.5$, which $\sigma$ should I put at the denominator of the definition of $t$?
While we're on this topic, I want to test my understanding of the $t$-test: once I have the $t$ value (let's say $t = 2.6$), if I want a 2-tailed test (like in the case of the absolute B-magnitudes of elliptical galaxies) I look in the tables or resolve numerically the integral of $t$ distribution with $\nu = N-1$ degrees of freedom between $-t$ and $+t$. In this example ($t = 2.6$, $\nu = 12$) I find a level of confidence of 97.7%, that is a level of significance (i.e. a $P$-value) of 2.3%. If my significance threshold was $\alpha = 5\%$ the null hypothesis is rejected, then I can say that the sample comes from the population; if instead it was $\alpha = 1\%$ the null hypothesis is not rejected, and I cannot prove it is false (but not even true).
 A: Welcome to CrossValidated. I'll try to answer your question to the best of my ability. I'll start off by perhaps restating our data to better understand the formula for our chosen $t$-test. 
From my understanding of your description, you've collected a sample of thirteen galaxies, $n = 13$ which have a mean absolute magnitude, $\bar{x}$, of $-20$ and a sample standard deviation, $S_x$ of $.5$.
Based upon previous research, the hypothesized population mean, $\mu$, is $-21.3$, with an unknown standard deviation, $\sigma$. 
Yyour hypothesis states that you wish to test whether your sample of galaxies was taken from (sampled from) the population. 

I want to conduct a t-test in order to know if this sample is taken
  from the population.

I should note that a $t$-test doesn't actually perform this hypothesis test. However, what it can do is test whether the stated population parameter, $\mu$ is likely to be equal to $-21.3$ based upon your sample. Therefore our hypotheses will be as follows:
$h_0, \mu = -21.3$,
$h_1, \mu \neq -21.3$
Since we're testing this hypothesis using one sample, this test is known as the one-sample $t$-test, which has the formula:
$$t = \frac{ \bar{x} - \mu }{S_\bar{x}} \, \mbox{,}$$
Where $S_\bar{x}$, is the standard error associated with the sample which has the formula:
$$S_\bar{x} = \frac{ S_x }{\sqrt{n}} \, \mbox{,}$$
So to answer your question

which standard deviation have I to put as denominator in t definition?

TL;DR: You should be using the sample standard error
