In the case measures, assume that we are considering a positive measure $\mu$ on $\mathcal{B}(R^{2})$-which is the set of borel sets in $R^{2}$, that has density f(x,y)-it can be seen as density of polulation. Then, when we consider a borel set $A\subset R^{2}$, we obtain the information of the population in the set A which is
$$\mu(A)=\int_{A} f(x,y)d\mu.$$
So, now could you give me some practical examples that can illustrate the concept of vector measures?.
I cannot tell you an application since this is not my area of expertise. However, I can explain the example that is commonly given such as Wikipedia (https://en.wikipedia.org/wiki/Vector_measure).
Normally a measure maps a set to a real number. However, vector measures allow you to instead map a set to something in a vector space. For example you can map it to the set of real numbers to all essentially bounded functions. One map that does this is the map that sends a set A to the characteristic function of A. This is a bounded function since the characteristic function is either 0 or 1. Each set will be mapped to a different characteristic function.
You could also use a similar map to map sets of the real numbers to the set $L^1$. This is the set of functions that have a finite integral. Note that for this example we would need to say $L^1(a,b)$ where an and b are finite. If we did not, then a $\sigma$-algebra containing $(n,n+1)$ for all $n \in \mathbb{N}$ would have to contain $\cup (n,n+1)$ and the characteristic function of $\cup (n,n+1)$ is not in $L^1(\mathbb{R})$. However, on a bounded interval the characteristic function will be integrable.
In summary, think of a vector measure as mapping a set to a vector instead of a set to a real number.