Initially RNAi was used to refer a process in which exogenous small(~22nt) RNAs (siRNA) silence expression of certain mRNAs which have regions complementary to the siRNA. This definition now also includes miRNA regulated pathway.
The main catalytic component of RNAi is an Ribo-endonuclease called Argonaute (Ago), which cleaves the target mRNAs. dsRNAs are cleaved by Dicer to give rise to siRNAs. Different species have different number of paralogous Argonaute proteins. In mammals only Ago2 has the endonucleolytic activity.
miRNAs are endogenous RNAs similar to siRNAs in function and biogenesis. However, miRNAs targets are usually localized in 3'UTR of mRNAs are are usually not perfectly complementary to the miRNAs. Moreover, the miRNA targets are not directly cleaved; degradation happens indirectly by decapping/deadenylation and subsequent exonucleolytic decay. Repression can also happen without degradation and in these cases the repressive effect is reversible.