A component is a subset of the network for which all pair of vertices have at least one path, and which is maximal (i.e no extra nodes can be added that preserve this property). A connected graph has only one component, while a disconnected one has more than one.

The adjacency matrix can always be written in block diagonal form with blocks corresponding to components.

Components in directed networks

Weakly connected components are components of a directed network ignoring the direction.

Strongly connected components have a path between any two vertices in both directions.

Acyclical graphs can't have strongly connected components with >1 vertex since, they would necessarily include a cycle.

Out-components are all the vertices reachable from a certain vertex, including the vertex itself.
In-components are all the vertices from which one can reach a certain vertex, including the vertex itself.

Both of these are identical for all vertices in a strongly connected component.