Hartshorne is the reference where you can find the following example which might be useful. I what follow everything is with multiplicity. Now Alberto pointed out above the case of the divisor over $\mathbb{P}^1$ associated to its "tangent bundle": Two points over the sphere counted with multiplicity (from here though, it is not hard to believe that the Chern class of such a bundle is going to be 2). Notice that these two points are given by zeros of polynomials of degree two defined over the sphere. I think nothing stops you taking now polynomial of degree 3, 4 and so on. Then what we get are nothing but 3, 4 points over the sphere: Divisors of degree 3, 4 and so on. We can do something similar over all the curves (Riemann Surfaces) and what we get are divisors: points with labels. Such labels are the multiplicities. Chapter IV Hartshorne. or Klaus-Hulek: Elementary Algebraic Geometry.

Now, let's take a look at divisors over the surface $\mathbb{P}^2$: they are algebraic curves (Riemann Surfaces). Do not get confused please by the name Surface here. Applying the same argument as before, a divisor of degree two is going to be the zero locus of polynomials of degree 2: conics. Same for degree three (cubics), four (quartics), and so on and so forth. For instance, in degree two we might have the divisor $C=([x:y:z]\in \mathbb{P}^2|\ \ x^2+y^2=z^2)$.

Now taking a family of divisors of degree two, the conics, it is well known that the space of embeddings of conics in $\mathbb{P}^2$ is (the linear system) $\mathbb{P}^5$. We get this by considering the coefficients in the equation $ax^2+by^2+cz^2+dxy+exz+fyz=0$ as coordinates in $\mathbb{P}^5$. Notice that we get the following map out of the previous considerations, $$\phi:\mathbb{P}^2\rightarrow \mathbb{P}^5$$ given by $[x:y:z]\mapsto [x^2:y^2:z^2:xy:xz:yz]$. Here pencils are a subfamily of conics in the complete linear system given above with a certain property (find out which one). However, we can consider the following subfamily of conics: all those conics passing through a fixed point in $\mathbb{P}^2$. This is nothing but a hyperplane $H$ in $\mathbb{P}^5$. We can even consider $\phi(\mathbb{P}^2)\cap H$. This is going to be a divisor in $\mathbb{P}^2\cong \phi(\mathbb{P}^2)$. Guess which one?. Hartshorne II section 7.

One can apply the the ideas with zero locus of polynomials of degree three: Divisors of degree 3 in $\mathbb{P}^2$. These have the name of elliptic curves. (did someone say that in considering these curves we find the divisor associated to the canonical bundle of $\mathbb{P}^2$?) We can go on with the degree and getting divisors on the projective plane of higher degree.

Hartshorne is the reference where you can find the following example which might be useful. I what follow everything is with multiplicity. Now Alberto pointed out above the case of the divisor over $\mathbb{P}^1$ associated to its "tangent bundle": Two points over the sphere counted with multiplicity (from here though, it is not hard to believe that the Chern class of such a bundle is going to be 2). Notice that these two points are given by zeros of polynomials of degree two defined over the sphere. I think nothing stops you taking now polynomial of degree 3, 4 and so on. Then what we get are nothing but 3, 4 points over the sphere: Divisors of degree 3, 4 and so on. We can do something similar over all the curves (Riemann Surfaces) and what we get are divisors: points with labels. Such labels are the multiplicities. Chapter IV Hartshorne. or Klaus-Hulek: Elementary Algebraic Geometry.

Now, let's take a look at divisors over the surface $\mathbb{P}^2$: they are algebraic curves (Riemann Surfaces). Do not get confused please by the name Surface here. Applying the same argument as before, a divisor of degree two is going to be the zero locus of polynomials of degree 2: conics. Same for degree three (cubics), four (quartics), and so on and so forth. For instance, in degree two we might have the divisor $C=([x:y:z]\in \mathbb{P}^2|\ \ x^2+y^2=z^2)$. Deshomogenizing with $H=[z=1]$ you get a perfect polynomial $x^2+y^2=1$ which defines the intersection $H\cap C$. This is how your global divisor $C$ looks like locally.

Now taking a family of divisors of degree two, the conics, it is well known that the space of embeddings of conics in $\mathbb{P}^2$ is (the linear system) $\mathbb{P}^5$. We get this by considering the coefficients in the equation $ax^2+by^2+cz^2+dxy+exz+fyz=0$ as coordinates in $\mathbb{P}^5$. Notice that we get the following map out of the previous considerations, $$\phi:\mathbb{P}^2\rightarrow \mathbb{P}^5$$ given by $[x:y:z]\mapsto [x^2:y^2:z^2:xy:xz:yz]$. Here pencils are a subfamily of conics in the complete linear system given above with a certain property (find out which one). However, we can consider the following subfamily of conics: all those conics passing through a fixed point in $\mathbb{P}^2$. This is nothing but a hyperplane $H$ in $\mathbb{P}^5$. We can even consider $\phi(\mathbb{P}^2)\cap H$. This is going to be a divisor on $\mathbb{P}^2\cong \phi(\mathbb{P}^2)$. Guess which one?. Hartshorne II section 7.

One can apply the the ideas with zero locus of polynomials of degree three: Divisors of degree 3 in $\mathbb{P}^2$. These were given the name of elliptic curves. (did someone say that in considering such curves, we find the divisor associated to the canonical bundle of $\mathbb{P}^2$?). We can go on with the degree and getting divisors on the projective plane of higher degree. These were only examples of divisors on $\mathbb{P}^2$. Notice that all of them have a nontrivial topology and geometry. This fact is not a coincidence and the book of HG argues in this direction in Chapter zero.