I'm new to the Measure Theory. I was wondering can we always find a measure for a measurable space? It would be better to explain in details.
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There's always the measure that maps every subset to $0$.

If you mean a probability measure then it also exists, pick $x\in X$ and define $f(A)=1$ if $x\in A$ and $0$ otherwise. This is called a Dirac measure.
To construct a probability measure on any measurable space $(\Omega,\tau)$, consider the Dirac measure P:
For any $A\in \tau$
$$P(A) = \begin{cases}
1 & \text{$A\ni x$}, \\
0 & \text{otherwise}.
\end{cases}$$
Choice $x\in \Omega$ such that: $\exists B\in \tau, B\ni x $. Then $(\Omega,\tau,P)$ is a probability space, P is a probability measure.