δ(Qi,x) = Qj where x is a string and it have only one input symbol,
δ*(Qi,x) = Qj where x is a string and it have more than input symbol, i.e., it means you will be reach Qj state after processing the string x.
δ*(Qi,"ABC") = δ ( δ( δ(Qi,"A") ,"B") ,"C")