Step 1: Concept
According to the Brønsted-Lowry acid-base theory, an acid turns into its conjugate base after donating a proton ($\text{H}^{+}$).
Step 2: Meaning
To determine the conjugate base of any chemical species, remove exactly one hydrogen ion ($\text{H}^{+}$) from its formula and decrease its net electrical charge by one unit.
Step 3: Analysis
Starting with the dihydrogen phosphate ion, $\text{H}_{2}\text{PO}_{4}^{-}$:
1. Remove one $\text{H}^{+}$ ion, leaving behind $\text{HPO}_{4}$.
2. Decreasing the charge from $-1$ by one more unit results in a charge of $-2$.
This yields the monohydrogen phosphate ion, $\text{HPO}_{4}^{2-}$.
Step 4: Conclusion
Thus, the conjugate base of $\text{H}_{2}\text{PO}_{4}^{-}$ is $\text{HPO}_{4}^{2-}$.
Final Answer: (A)