At that moment – at that particular time.

• I was just about to enter the interview room. At that moment an old friend of mine appeared like a bolt from the blue.
• I was just reaching the railway station. Just at that moment, my car broke down.

At the moment – now, at the present time.

• At the moment I’m working in a restaurant.
• At the moment I’m reading an interesting novel. (Not necessarily one is reading the novel right now, but he has started to read the novel and covered a portion of it, but not finished)

The moment – as soon as.

• The moment I saw him, I recognized him.
• The moment she received the letter, she replied to it.
• The moment Kerala was in the grip of the flood, thousands of fishermen jumped into rescue action risking their lives.