Let $U^{ n}$ be the unit polydisc of ${ \Bbb C}^{ n}$ and $\phi=(\phi_1, >..., \phi_n)$ a holomorphic self-map of $U^{ n}.$ By ${ \cal B}^p(U^{ n})$, ${ \cal B}^p_{ 0}(U^{ n})$ and ${ \cal B}^p_{ 0*}(U^{ n})$ denote the $p$-Bloch space, Little $p$-Bloch space and Little star $p$-Bloch space in the unit polydisc $U^n$ respectively, where $p, q>0$. This paper gives the estimates of the essential norms of bounded composition operators $C_{ \phi}$ induced by $\phi$ between ${ \cal B}^p(U^n)$ (${ \cal B}^p_{ 0}(U^n)$ or ${ \cal B}^p_{ 0*}(U^n)$) and ${ \cal B}^q(U^n)$ (${ \cal B}^q_{ 0}(U^n)$ or ${ \cal B}^q_{ 0*}(U^n)$). As their applications, some necessary and sufficient conditions for the bounded composition operators $C_{ \phi}$ to be compact from ${ \cal B}^p(U^n)$ $({ \cal B}^p_{ 0}(U^n)$ or ${ \cal B}^p_{ 0*}(U^n))$ into ${ \cal B}^q(U^n)$ (${ \cal B}^q_{ 0}(U^n)$ or ${ \cal B}^q_{ 0*}(U^n)$) are obtained.
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