Intuition of critical values in t-distribution below $N=50$ or below sufficiently large $N$ 
I understand how the z-scores are defined in the standard normal
distribution and I also understand how when $N$ extends to infinity
why the t-distribution approaches and equals the standard normal
distribution.

My question: How are the values in the statistical table for the t-distribution calculated/defined for low values of $N$, because they seem arbitrary to me?
 A: Suppose $n = 10.$ One can compute many t statistics from normal samples of size $n$ where the null hypothesis $H_0: \mu-0$ is true in order
to get a good idea of the shape of Student's t distribution
with $\nu = n-1 = 9$ degrees of freedom. By finding quantile $0.975$
of the results one can find the upper critical value for a two-sided test at the 5% level. [The R code below generates a million normal samples and finds their corresponding t statistics.
With a million samples one can expect about 2-place accuracy.]
set.seed(1031)
t = replicate(10^6, t.test(rnorm(10))$stat)
quantile(t, .975)
   97.5% 
2.256101       # aprx 2.262
qt(.975, 9)
[1] 2.262157   # exact value

You can find the exact value of quantile 0.975 of $\mathsf{T}(\nu=9);$ look on line 9 of a printed t table. By symmetry,
the lower critical value for a two-sided test at level 5% is
$-2.262.$
hdr = "Simulated Dist'n of T(9) with PDF"
hist(t, prob=T, br=50, ylim=c(0,.5), col="skyblue2", main=hdr)
curve(dt(x,9), add=T, col="orange", lwd=2)
abline(v = 2.262, col="blue", lty="dotted")


Below is a plot of the density functions of $\mathsf{T}(\nu=0)$ and
$\mathsf{Norm}(0,1).$ They are very nearly the same.
However, they are sufficiently different that $2.262$ cuts
2.5% of the probability from the upper tail of the t distribution, while $1.96$ cuts 2.5% from the upper tail of the standard normal distribution. [Respective standard deviations are $\sqrt{9/7} = 1.1339$ and
$1.]$

R code for figure just above:
hdr="PDFs of T(9) [orange] and NORM(0,1) [dashed]"
 curve(dt(x,9), -5, 5, col="orange", lwd=2, main=hdr)
 curve(dnorm(x), add=T, lty="dashed")
 abline(h=0, col="green2"); abline(v=0, col="green2")

