Step 1: Recall how a peptide bond forms. A peptide bond joins two amino acids by linking the carboxyl group of one to the amino group of the other. Step 2: Identify the reacting groups. One amino acid offers its $-COOH$ group and the other offers its $-NH_2$ group for the reaction. Step 3: See which atoms are removed. The $-OH$ leaves from the carboxyl group and an $-H$ leaves from the amino group. These two pieces combine. Step 4: Form the by-product. An $-OH$ plus an $-H$ together make $H_2O$, so a water molecule is eliminated during the condensation. Step 5: Write the reaction. \[ R\text{-}COOH + H_2N\text{-}R' \rightarrow R\text{-}CONH\text{-}R' + H_2O \] where $-CONH-$ is the peptide linkage. Step 6: Reject the other choices. $NH_3$, $CH_3OH$ and $CO_2$ are not released here, so the eliminated molecule is water, option 1. \[ \boxed{H_2O} \]