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.