Step 1: Concept
According to Br\o{}nsted--Lowry theory, an acid donates a proton ($H^+$) and a base accepts a proton. Each acid--base pair is related by the gain or loss of one proton.
Step 2: Meaning
The conjugate pairs are: $\text{Acid}_1 / \text{Base}_2$ and $\text{Base}_1 / \text{Acid}_2$.
Step 3: Analysis
In the given equation:
$HCl$ donates $H^+$ to become $Cl^-$ $\Rightarrow$ $HCl = \text{Acid}_1$, $Cl^- = \text{Base}_2$.
$H_2O$ accepts $H^+$ to become $H_3O^+$ $\Rightarrow$ $H_2O = \text{Base}_1$, $H_3O^+ = \text{Acid}_2$.
Step 4: Conclusion
$\text{Base}_2$ is the conjugate base of $HCl$, which is $Cl^-$.
Final Answer: (C)