The arabic dua after adhan with an english transalation mentioned in a hadeeth from Sahih al-Bukhari.

عَنْ جَابِرِ بْنِ عَبْدِ اللهِ أَنَّ رَسُولَ اللهِ صلى الله عليه وسلم قَالَ : مَنْ قَالَ حِينَ يَسْمَعُ النِّدَاءَ

اللَّهُمَّ رَبَّ هَذِهِ الدَّعْوَةِ التَّامَّةِ وَالصَّلاَةِ الْقَائِمَةِ آتِ مُحَمَّدَانِ الْوَسِيلَةَ وَالْفَضِيلَةَ وَابْعَثْهُ مَقَامًا مَحْمُودَانِ الَّذِي وَعَدْتَهُ

حَلَّتْ لَهُ شَفَاعَتِي يَوْمَ الْقِيَامَةِ.
(رواه البخاري، باب الدعاء عند النداء، ح/٦١٤)
 

Jabir ibn Abdullah narrates that the messenger of Allah, Allah send peace and salutations upon him said, whoever recites the following when he hears the call (to prayer, i.e. after the Adhan) then my intercession for him becomes binding on the day of resurrection.

Oh Allah! Lord of this perfect call and this prayer to be established, grant Muhammad Al-wasilah (a high and special place in Jannah) and Al-fadheelah (a rank above the rest of creation) and raise him to a praised platform which you have promised him.

The complete method of reciting this Dua is explained other narrations.

Amr Ibn Al-as narrates that he heard the prophet, Allah send peace and salutations upon him saying, ”When you hear the Muadhin (caller to prayer) then say as he says, then send salutations upon me (once the adhan is finished, as during the Adhan one is replying to the Adhan itself), as the the one who asks Allah to send peace and salutations upon me, Allah sends mercy ten mercies upon him. Then ask Allah to grant me Al-wasilah, a place in Jannah, which is only befitting for one slave of Allah, and I hope that I am that one slave. Whoever asks Allah to grant me Al-Wasilah, (my) intercession (for him) becomes binding. (Muslim)

In another narration, also by Imam Muslim the messenger of Allah, Allah send peace and blessings upon him advised that the reply to the Muadhin (caller to prayer) be given in the following way with the words recited not only by the mouth, but by the heart as well and whoever does so will enter Jannah. (Muslim)

Muadhin: اللَّهُ أَكْبَرُ اللَّهُ أَكْبَرُ
Reply: اللَّهُ أَكْبَرُ اللَّهُ أَكْبَرُ

Muadhin: أَشْهَدُ أَنْ لاَ إِلَهَ إِلاَّ اللَّهُ
Reply: أَشْهَدُ أَنْ لاَ إِلَهَ إِلاَّ اللَّهُ

Muadhin: أَشْهَدُ أَنَّ مُحَمَّدًا رَسُولُ اللَّه
Reply: أَشْهَدُ أَنَّ مُحَمَّدًا رَسُولُ اللَّه

Muadhin: حَىَّ عَلَى الصَّلاَةِ / حَىَّ عَلَى الْفَلاَحِ
Reply: لاَ حَوْلَ وَلاَ قُوَّةَ إِلاَّ بِاللَّه

Combining the above three narrations, we learn that one should reply to the adhan when the adhan is taking place, after which he should send salutaions upon the messenger and then recite the Dua.

In another narration, also narrated by Imam Muslim, the Prophet, Allah send peace and salutations upon him encouraged to recite the following, and whoever does so, Allah will forgive all his minor sins.

 
أَشْهَد أَنْ لَا إِلَه إِلَّا اللَّه وَحْده لَا شَرِيك لَهُ ، وَأَنَّ مُحَمَّدًا عَبْده وَرَسُوله ، رَضِيت بِاَللَّهِ رَبًّا وَبِمُحَمَّدٍ رَسُولًا ، وَبِالْإِسْلَامِ دِينًا

I bear witness that there is none worthy of worship besides Allah alone, He has no partner, and I bear witness that Muhammad is His slave and messenger. I am pleased with Allah as his lord (and only worships Him), and Muhammad as his messenger (I accets him to be the best example) and Islam as his religion (I accept it to be the best way of life).

This may be recited after the Muadhin says:

أَشْهَد أَنْ لَا إِلَه إِلَّا اللَّه

Or after the Adhan.