This paper discusses new error bounds for the tensor complementarity problem using a $P$-tensor. A new lower error bound and a global error bound are presented for such a problem. It is proved that the norm of the exact solution of the tensor complementarity problem with a $P$-tensor has a lower bound and an upper bound. When the order of a tensor is $2$, all the results for the tensor complementarity problem obtained reduce to those for the linear complementarity problem.