Consider a directed graph given in below, DFS of the below graph is 1 2 4 6 3 5 7 8. In below diagram if DFS is applied on this graph a tree is obtained which is connected using green edges.
 

Tree Edge: It is an edge which is present in the tree obtained after applying DFS on the graph. All the Green edges are tree edges. 

Forward Edge: It is an edge (u, v) such that v is descendant but not part of the DFS tree. Edge from 1 to 8 is a forward edge. 

Back edge: It is an edge (u, v) such that v is ancestor of node u but not part of DFS tree. Edge from 6 to 2 is a back edge. Presence of back edge indicates a cycle in directed graph. 

Cross Edge: It is a edge which connects two node such that they do not have any ancestor and a descendant relationship between them. Edge from node 5 to 4 is cross edge.
 

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