Concept:
Different protozoan parasites cause different diseases in humans. Identification of the disease caused by a pathogen is an important topic from Human Health and Disease.
Step 1: Identify the organism given in the question.
The organism mentioned is
\[
\textit{Leishmania donovani}
\]
which is a flagellated protozoan parasite.
Step 2: Recall the disease caused by this parasite.
\[
\textit{Leishmania donovani}
\longrightarrow
\text{Kala-azar}
\]
Kala-azar is also called
\[
\text{Visceral Leishmaniasis}
\]
Step 3: Check the remaining options.
\[
\begin{aligned}
\text{Delhi boils} &\rightarrow \textit{Leishmania tropica}\\
\text{Elephantiasis} &\rightarrow \textit{Wuchereria bancrofti}\\
\text{Oriental sores} &\rightarrow \textit{Leishmania tropica}
\end{aligned}
\]
Therefore these options are incorrect.
Step 4: Select the correct option.
Hence,
\[
\boxed{\text{Kala-azar}}
\]
is the disease caused by
\[
\textit{Leishmania\ donovani}
\]
Therefore,
\[
\boxed{\text{Option (A)}}
\]
is correct.