A Subset S of vertices of a Graph G is called a vertex cover if S includes at least one end point of every edge of the Graph. A Vertex cover S of G is a connected vertex cover if the induced subgraph of S is connected. The minimum cardinality of such a set is called the connected vertex covering number and it is denoted by $\alpha_c(G)$. A Vertex cover S of G is a total vertex cover if the induced subgraph of S has no isolates. The minimum cardinality of such a set is called the total vertex covering number and it is denoted by $\alpha_t(G)$. In this paper a few properties of connected vertex cover and total vertex covers are studied and specific values of $\alpha_c(G)$ and $\alpha_t(G)$ of some well-known graphs are evaluated.