Before answering the question of where does a peptide bond form, I would like to say that protein is synthesized in the ribosome located in the cytosol. A ribosome is a protein (practically polypeptide) synthesizing apparatus where matured mRNA (after processing) comes and genetic information in the form of genetic codons is translated into the polypeptide. The ribosomes are made of two different subunits. In the case of prokaryotes, the bigger subunit of the ribosome is the 50S and the smaller subunit is the 30S whereas in the case of eukaryotes 60S and 40S are the bigger and smaller subunits respectively.
Now, I will come to the point where does a peptide bond form? At first, the initiation codon (AUG) of the mRNA is recognized by a signal sequence located in the 16S ribosomal rRNA of the 30S subunit of the ribosome that promotes the binding of the mRNA to the 30S subunit. Binding of the mRNA chain to the 30S subunit is promoted by the initiation factors IR1 and IR3. After binding of mRNA to the 30S subunit, an amino acid carrying tRNA which is also called as charged tRNA (first amino acid to be incorporated into a polypeptide chain; formyl-methionine in prokaryotes and methionine in eukaryotes) along with initiation factor IF2 binds to the P site first codon of the mRNA.
All those initiation factors; IR1, IR2, and IR3 are released when the 50S subunit binds to the mRNA-30S subunit complex forming a closed 70S complex. 30S subunit has only two sites, A site, and P site while the 50S subunit has three sites; A site, P site, and E site. If you are curious about the name of the sites then I will make it clear. A site means an aminoacyl site, P site means peptidyl site, and E site means exit site.
So, it is the P site or peptidyl site where a peptide bond is formed. At first, charged tRNA carrying the first amino acid to be incorporated is placed into the P site and the second charged tRNA is placed into the A-site. In the p site, a peptide bond is formed catalyzed by peptidyl transferase activity of the 23S rRNA (ribozyme) of the 50S subunit of the ribosome. Once peptide bond is formed, the first tRNA carrying no any amino acid is translocated to the E site from where it is released out of the complex, while dipeptidyl group carrying tRNA is translocated from A site to P site catalyzed by translocase leaving the A site for another incoming charged tRNA.
Tn the same way, the Elongation process proceeds with the help of elongation factors and elongation of the polypeptide chain continues until the stop codon arrives at the A site. Once a stop codon arrives at the A site, the release factor binds to the stop codon and prevents binding of any charged tRNA to the A site and the polypeptide chain is released from the ribosome complex. The release of the polypeptide from the tRNA is mediated hydrolytic activity of the peptidyl transferase ribozyme and the ribosome complex itself is disassembled into its subunits. I hope now you know where does a peptide bond form.