Let $D$ be a co-complete category and $C$ be a small category. For a functor $F:C^{op}\times C \to D$ one defines the co-end
$$
\int^{c\in C} F(c,c)
$$
as the co-equalizer of
$$
\coprod_{c\to c'}F(c,c'){\longrightarrow\atop\longrightarrow}\coprod_{c\in C}F(c,c).
$$
It is the indexed co-limit $\mbox{colim}_W F$ where the weight is the functor $W:C^{op}\times C \to Set$ given by $Hom(-,-)$.

I have two strongly related questions regarding this definition. First, what's the intuition behind this construction? Can I think of it as a kind of ''fattened'' colimit? Second, why is the integral sign used for this? Can ordinary integration be related to this construction?

Martin's comment is right on the money; in particular, the best way to get a feeling for coends is through the many examples where they appear, such as (generalized) tensor products. But from an abstract point of view, coends can be considered as universal "extranatural transformations", and the ubiquity of coends (and ends) is explained by the ubiquity of such extranatural transformations. 

Let me take a specific example before getting into the abstract aspects. Let's consider the example of geometric realization of simplicial sets, as a left adjoint to the singularization functor $S: Top \to Set^{\Delta^{op}}$. Recall that if $Y$ is a space, then $S(Y)$ is the simplicial set $\Delta^{op} \to Set$ whose value at the ordinal $[n]$ (with $n+1$ points) is defined by 

$$S(Y)([n]) = \hom_{Top}(\sigma_n, Y)$$ 

where $\sigma_n$ is the $n$-dimensional affine simplex seen as a topological space. We are interested in constructing a left adjoint $R$ to $S$, so that for any simplicial set $X$, the set of natural transformations 

$$X \to S(Y)$$ 

is in natural bijective correspondence with continuous maps $R(X) \to Y$. 

The way to do this is to "bend" a natural transformation 

$$X([n]) \to \hom_{Top}(\sigma_n, Y)$$ 

(a family of maps natural in $[n] \in \Delta$) into another family 

$$\phi_n: X([n]) \times \sigma_n \to Y$$ 

of continuous maps indexed by $n$. This family has a property intimately related to the naturality of the first family; it is called "extranaturality". It means that given any morphism $f: [m] \to [n]$, the composite 

$$X([n]) \times \sigma_m \stackrel{X[f] \times id}{\to} X([m]) \times \sigma_m \stackrel{\phi_m}{\to} Y$$ 

equals the morphism 

$$X([n]) \times \sigma_m \stackrel{id \times \sigma_f}{\to} X([n]) \times \sigma_n \stackrel{\phi_n}{\to} Y;$$ 

this precisely mirrors the naturality of the first family in $n$. Thus, what we are after is an extranatural transformation 

$$X([n]) \times \sigma_n \to R(X)$$ 

with the universal property that given any extranatural transformation $\phi_n$ as above, there exists a unique map $R(X) \to Y$ making the evident triangle commute (for each $n$). This is of course the coend 

$$R(X) = \int^n X([n]) \times \sigma_n$$ 

and the appropriate construction in terms of coproducts and coequalizers that you indicated in your question is exactly what is required to construct the universal extranatural transformation. 

This is easily abstracted. Given any functor $F: C^{op} \times C \to D$, one can define what it means for a family of maps $F(c, c) \to d$ (for fixed $d$) to be extranatural in $c$, and the coend is again described as a universal extranatural transformation. In nature, such transformations almost invariably arise by "bending" a natural transformation into an extranatural (also called "dinatural") one. The tensor product mentioned by Martin fits into this pattern: thinking of a left $R$-module map of the form 

$$M \to \hom_{Ab}(N, A)$$ 

($M$ a left $R$-module, $N$ a right $R$-module, $A$ an abelian group; the hom acquires a left $R$-module structure) as a $Ab$-enriched natural transformation between functors of the form $R \to Ab$ (where a ring $R$ is viewed as a one-object $Ab$-enriched category), we can "bend" this map into a map 

$$M \otimes N \to A$$ 

which is extranatural with respect to scalar actions, and the quotient $M \otimes N \to M \otimes_R N$ is the universal such extranatural map. But this only scratches the surface of possibilities for this type of situation. 

Finally, I second Martin's remark on the traditional integral notation -- not too much should be made of this, except that weighted colimits are primary examples of coends, and there are interchange isomorphisms which are reminiscent of Fubini theorems; this is touched upon in Categories for the Working Mathematician. 
