Graphs - Matrix Representation

Adjacency Matrix - One row and one column for each vertex. A one in the graph indicates that the vertices are adjacent. i.e - if there is a one in the A,A column, that means there is a loop. Because there are only 1's and 0's in the matrix, it is a Boolean matrix. Unfortunately it doesn't tell us how many edges are between any vertex.
Incidence Matrix - Has a row for each Vertex, and a column for each Edge. A one indicates that there is an end point for an edge - generally there will be two 1's for each edge, unless it represents a loop.