What is the prayer of the Quran for the deceased?

There are several prayers for the deceased in the Quran; I will list a few here because reciting the Quran is the best method to become familiar with them.

Arabic

ربنا اغفر لنا ولإخواننا الذين سبقونا بالايمان ولا تجعل في قلوبنا غلا للذين امنوا  ربنا إنك رؤوف رحيم Surah Al-Hashr Ayah 10

Transliteration

Rabana Ighfir Lana wa Li ikhwanina Allazina Sabaqona bi Al-Iman wa la Tag’al fi Qolobina Ghillan lil-Athina Amanu, Rabana Inka Ra’ouf Rahim.

English

Our Lord, forgive us and forgive our brothers who have

Arabic

ربنا اغفر لي ولوالدي وللمؤمنين يوم يقوم الحساب Surah Ibrahim Ayah 41

Transliteration

Rabana Ighfir Li wa li-walidaiia wa lil mu’minen yawma yagowm Al-Hisab 

English

Our lord forgive me, my parents and the believers on the day of judgment

Observe how, as Muslims, we offer du'a for all deceased Muslims, not only our departed parents or relatives. This is a link in the chain that unites the whole Muslim Ummah.

Which Tasbeeh should I read after a death?

You can utter many tasbeeh in memory of a deceased person. The most advised is (Istigfar), which entails pleading with Allah for their pardon and generally reciting duas to ensure their comfort in the afterlife, admission into paradise, protection from the fires of Hell, and other blessings.

Examples:

Arabic:

اللهم اغسله بالماء والثلج والبرد ونقه من الخطايا كما ينقى الثوب الأبيض من الدنس

Transliteration:

Allahuma Ighsilhu Bi Al-ma’a wa Al-thalg wa Al-Barad, wa Naqqihi mina al-khataya kama Unaqqa al-thawb al-abiad mina aa-danas

English:

O Allah, Wash him with water, and ice and cleanse him from sins as a white dress is cleansed from dirt

Arabic:

اللهم اغفر له وارحمه واعف عنه وعافه وأكرم نزله ووسع مدخله وثبته عند السؤال

Transliteration:

Allahuma Ighfir Lahu wa irhamhu, wa I’fu anhu, wa Akrim Nuzulahu, wa wasi’ mudkhalahu, wa thabitthu inda al-suo’al

English:

O Allah, forgive him, have mercy on him, pardon him, make him stay in a good place, make his entrance wide, and keep him steady while being questioned

Arabic:

اللهم ادخله الجنة واعذه من عذاب القبر وعذاب النار

Transliteration:

Allahuma Adkhilu al-Jannah wa A’izhu min ‘Azab Al-Qabr wa ‘Azab Al-Nar 

English:

O Allah, grant him paradise and protect him from the tortures of the grave and the tourturs of the fire. 

Arabic:

اللهم أبدله داراً خيراً من داره وأهلاً خيراً من أهله وزوجاً خيراً من زوجه

Transliteration:

Allahuma Abdilhu Daran Khiran min Darhi wa Ahlan Khiran min Ahlih wa Zawjan Khairan min Zawjih

English:

O Allah, replace for him, a house that’s better than his house, a family that’s better than his family and a wife that’s better than his wife. 

Keep in mind that you can recite these dua anywhere, as Allah is all-knowing and all-hearing and the dead do not hear you pray or fast.