It's used in a similar way to English, to express:
* Understanding: "Aha! Now I get it." (¡Ajá! Ahora lo entiendo.)
* Surprise or realization: "Aha! You're right." (¡Ajá! Tienes razón.)
* Agreement or confirmation: "Aha, I knew it." (¡Ajá! Yo lo sabía.)
It can also be used as a slight expression of annoyance when someone is trying to be sneaky or when you feel they are trying to hide something from you.
Keep in mind that the pronunciation of "¡Ajá!" is with a strong emphasis on the "á."