All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
The Prophet, sallallaahu ‘alayhi wa sallam, said: "Between every two Athaans (i.e. Athaan and Iqaamah) there is a prayer, between every two Athaans, there is a prayer between every two Athaans," and after saying it for the third time he said: "for whoever wants to." [Al-Bukhari] It means that there is a prayer between the Athaan and the Iqaamah. The Prophet called the Athaan and the Iqaamah (second call to prayer) "two Athaans" because of their inseparability, exactly as the Arabs call the moon and the sun "two moons". Prayer in the Hadeeth refers to the voluntary prayer. So, it is permissible for the Muslim to perform whatever number of voluntary prayers between the Athaan and Iqaamah, except for the cases excluded. For instance, the Muslim should only perform two Rak‘ahs (units) before Fajr prayer. That is why the word prayer came in the Hadeeth with an indefinite article in order to include whatever number of prayers the person intends to offer.
The last phrase in the Hadeeth signifies that the Prophet, sallallaahu ‘alayhi wa sallam, after uttering the same statement for the third time, added: "for whoever wants to." This is to alleviate hardship and the obligation that might be inferred from the repetition of the order: "Between every two Athaans there is a prayer". By the "third", it does not mean that there is a new prayer as the questioner misunderstood.
As far as intention for voluntary prayers is concerned, voluntary prayers are divided into two types:
First: Specified voluntary prayers which are performed in certain occasions such as the eclipse prayer, rain-invoking prayer, Taraaweeh (voluntary prayer in Ramadan), Witr, and confirmed voluntary prayer – they all need an intention that the Muslim, for instance, intends to perform a voluntary prayer for Thuhr or Maghrib prayer.
Second: General voluntary prayers, such as night prayers, which do not require a specific intention, only the intention of performing prayer.
Allaah Knows best.