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.
A *measurable space* is just a pair $(X,\mathcal{M})$, where $X$ is a set and $\mathcal{M}\subseteq\mathscr{P}(X)$ is a $\sigma$-algebra on $X$. The purpose of this definition is just to identify a $\sigma$-algebra on $X$, which is the collection of all measurable subsets of $X$.

For any given set $X$, there are several ways in which we can define a measure $\mu:\mathscr{P}(X)\to[0,\infty]$. (See examples below.) For any such measure, the restriction $\mu|\_{\mathcal{M}}$ will turn the measurable space $(X,\mathcal{M})$ into the measure space $(X,\mathcal{M},\mu|\_{\mathcal{M}})$.
In fact, it's worth proving the following easy exercise.

> 
> If $(X,\mathcal{M},\mu)$ is a measure space and $\mathcal{N}\subseteq \mathcal{M}$ is a $\sigma$-algebra, then $\mu|\_{\mathcal{N}}$ is a measure on the measurable space $(X,\mathcal{N})$.
> 
> 
> 

Some important examples of measures $\mu:\mathscr{P}(X)\to[0,\infty]$:

1. The zero measure: $\mu(A)=0$ for every $A\subseteq X$.
2. The infinite measure: $\mu(A)=\infty$ for $A\ne\emptyset$ and $\mu(\emptyset)=0$.
3. The counting measure:
$$
\mu(A) = \begin{cases}
|A| & \text{if $A$ is finite}, \\
\infty & \text{otherwise}.
\end{cases}
$$
4. Point mass measure (or Dirac measure) of a fixed point $x\in X$:
$$\mu(A) = \begin{cases}
1 & \text{if $x\in A$}, \\
0 & \text{otherwise}.
\end{cases}$$
5. $\mu(A)=\infty$ if $A$ is infinite, and $\mu(A)=0$ otherwise.