The Laplacian matrix is the representation of a graph in matrix form.

Consider a simple graph $G$. Given it's degree matrix $D$, and adjacency matrix $A$, the Laplacian is defined as

$$L = D - A$$

With all matrices having size $|G| \times |G|$

The degree matrix $D$ consists only of entries in the diagonal, with $D_{i,i}$ representing the degree of node $i$.
The adjacency has entries $A_{i,j} = 1$ when there is an edge in $G$ from $i$ to $j$. These entries could also be used to represent edge weight in the case of a weighted graph, or the number of edges in the case of a multigraph. If the graph is undirected, the matrix will be symmetric.

The eigenvalues and eigenvectors of the Laplacian provide a rich amount of information about the properties of a graph, this is the field of Spectral Graph Theory.
In particular, the number of connected components in the graph is equal to the multiplicity of the 0 eigenvalue. The second smallest eigenvalue (also called Fiedler Value) is useful for graph partitioning and linear arrangement problems.