What to say upon hearing the Athan/Azan (call to prayer)  
 

22. Repeat what the Mu'aththin says, except for when he says:

"حَيَّ عَلَى الصَّلاَةِ وَحَيَّ عَلَى الْفَلَاحِ"

Hayya 'alas-Salaah (hasten to the prayer) and Hayya 'alal-Falaah (hasten to salvation). Here you should say:

"لَا حَوْلَ وَلَا قُوَّةَ إِلَّا بِاللهِ"

Laa hawla wa laa quwwata 'illaa billaah.

There is no might and no power except by Allah.

Reference: Al-Bukhari 1/152, Muslim 1/288.
 

"وَأَنَا أَشَْهَدُ أَنْ لَا إِلَهَ إِلَّا اللهُ وَحْدَهُ لَا شَرِيكَ لَهُ وَأَنَّ مُحَمَّداً عَبْدُهُ وَرَسُولُهُ، رَضِيتُ بِاللهِ رَبَّاً وَبِمُحَمَّدٍ رِسُولاً وَبِالإِْسلاَمِ دِيناً".

23. Wa 'anaa 'ash-hadu 'an laa 'ilaaha 'illallaahu wahdahu laa shareeka lahu wa 'anna Muhammadan 'abduhu wa Rasooluhu, radheetu billaahi Rabban, wa bi-Muhammadin Rasoolan wa bil'islaami deenan.

I bear witness that none has the right to be worshipped but Allah alone, Who has no partner, and that Muhammad is His slave and His Messenger. I am pleased with Allah as my Lord, with Muhammad as my Messenger and with Islam as my religion.¹ [To be recited in Arabic after the Mu'aththin’s Tashahhud or the words of affirmation of Faith] ²

Reference:
¹ Muslim 1/290.
²Ibn Khuzaymah 1/220.
 

"اللَّهُمَّ رَبَّ هَذِهِ الدَّعْوَةِ التَّامَّةِ، وَالصَّلَاةِ الْقَائِمَةِ، آتِ مُحَمَّداً الْوَسِيلَةَ وَالْفَضِيلَةَ، وَابْعَثْهُ مَقَاماً مَحْمُوداً الَّذِي وَعَدْتَهُ، [إَنَّكَ لَا تُخْلِفُ الْمِيعَادَ].

24,25. Allaahumma Rabba haathihid-da 'watit-taammati wassalaatil-qaa'imati, 'aati Muhammadanil-waseelata walfadheelata, wab 'ath-hu maqaamam-mahmoodanil-lathee wa'adtahu, ['innaka laa tukhliful-mee'aad]

After replying to the call of Mu'aththin. you should recite in Arabic Allah's blessings on the Prophet.

O Allah , Lord of this perfect call and established prayer. Grant Muhammad the intercession and favor, and raise him to the honored station You have promised him, [verily You do not neglect promises].

Reference:
Al-Bukhari 1/152, and the addition between brackets is from Al-Bayhaqi 1/410 with a good (Hasan) chain of narration. See 'Abdul-Azlz bin Baz's Tuhfatul-'Akhyar, pg. 38.
 

Between the call to prayer and the 'Iqamah, you should supplicate Allah for yourself. Invocation during this time is not rejected.