You have specified the null hypothesis that $X$ is a standard normal random variable. The alternative hypothesis is sometimes not specified explicitly,
though in simple examples of this kind, the alternative could be that $X$ is a
unit-variance normal random variable with mean $\mu \neq 0$.
A typical question that needs to be resolved is:
Given that we observed that $X$ has value $\alpha$,
is this observation consistent with the null hypothesis?
The idea here is that a standard normal random variable $X$
is quite unlikely to take on large positive or large
negative values. With high probability, $X$ lies in the
interval $[-3,+3]$. So if we had observed $X = 10$, say,
we could quite confidently reject the null hypothesis
since the alternative, that the observation came from
a distribution with mean $\mu$ closer to $10$ looks
to a more reasonable assumption. But even in the absence
of a specified (or vaguely specified or unspecified) alternative
hypothesis, the observation $X=10$ seems not very consistent
with the null hypothesis. This observation could occur
by chance even when the null hypothesis is true, but it is
our fondest hope we
hope that we have not been so unlucky when we confidently
reject the null hypothesis.
On the other hand, if $X = 0.1$, we would not be inclined
to reject the null hypothesis. It is perfectly consistent
with $X$ being a standard normal variable. But understand
that
not rejecting the null hypothesis is not the same
as a whole-hearted embrace or acceptance of the null
hypothesis.
All you are saying when you fail to reject the null
is that the available evidence is not strong enough to
force you into consideration of alternatives. Notice,
for example, that the observation $X=0.1$ is also
quite consistent with the hypothesis that $X$ is
a unit-variance normal random variable with mean
$0.00000001$, say, rather than the mean $0$ insisted
upon in the null hypothesis.
Now, turning to your specific problem,
$P\{|X| > 1.96\} = 0.05$ and so if you observe
that the observed value $\alpha$ of $X$ is outside
the interval $[-1.96,+1.96]$, you reject the null
hypothesis, while if $\alpha \in [-1.96,+1.96]$,
you do not reject the null hypothesis.
Your confidence level in this choice is
$0.95$.