Sajdah Al-Tilaawah is the prostration (sajdah) that one has to perform when one recites or hears, in the prayer or outside specific ayats containing words of sujood in the Quran.
Sujood al-tilaawah, whether during prayer or outside of it, is Sunnah, not obligatory. It is Waajib on both the reciter and all those who hear the recitation. There are fifteen ayats in the Quran where we should perform sajdah al-tilaawah (prostration of recitation) when reciting them.
Method of Performing Sajdah al-Tilaawah: All the Sharaait (preconditions) of Salaah other than Takbeer al-Tahreemah - i.e. cleanliness, facing the Qiblah, intention, covering body and time, must be satisfied.
Ibrahim b. Yazid al-Tayml reported: I used to read the Qur'an with my father in the vestibule (before the door of the mosque). When I recited the ayats concerning prostration, he prostrated himself. I said to him: Father, do you prostrate yourself in the path? He said: I heard Abu Dharr saying: I asked the Messenger of Allah (ﷺ) about the mosque that was first set up on the earth. He said: Masjid Harim. I said: Then which next? He said: The Masjid al-Aqsa. I said: How long is the space of time between the two? He said: Forty years. He (then) further said: The earth is a mosque for you, so wherever you are at the time of prayer, pray there.
حَدَّثَنِي عَلِيُّ بْنُ حُجْرٍ السَّعْدِيُّ، أَخْبَرَنَا عَلِيُّ بْنُ مُسْهِرٍ، حَدَّثَنَا الأَعْمَشُ، عَنْ إِبْرَاهِيمَ بْنِ يَزِيدَ التَّيْمِيِّ، قَالَ كُنْتُ أَقْرَأُ عَلَى أَبِي الْقُرْآنَ فِي السُّدَّةِ فَإِذَا قَرَأْتُ السَّجْدَةَ سَجَدَ فَقُلْتُ لَهُ يَا أَبَتِ أَتَسْجُدُ فِي الطَّرِيقِ قَالَ إِنِّي سَمِعْتُ أَبَا ذَرٍّ يَقُولُ سَأَلْتُ رَسُولَ اللَّهِ صلى الله عليه وسلم عَنْ أَوَّلِ مَسْجِدٍ وُضِعَ فِي الأَرْضِ قَالَ " الْمَسْجِدُ الْحَرَامُ " . قُلْتُ ثُمَّ أَىٌّ قَالَ " الْمَسْجِدُ الأَقْصَى " . قُلْتُ كَمْ بَيْنَهُمَا قَالَ " أَرْبَعُونَ عَامًا ثُمَّ الأَرْضُ لَكَ مَسْجِدٌ فَحَيْثُمَا أَدْرَكَتْكَ الصَّلاَةُ فَصَلِّ " .
Reference: Sahih Muslim 520 b; Book 5, Hadith 2; Book 4, Hadith 1057
1. outside Salaah
The method is for one to stand up, face Qiblah, say Allahu Akbar and go into Sajdah, pray Subhaana Rabbi'al A’laa’ (at least) 3 times while in Sajdah and then saying Allahu Akbar stand back up.
Aisha (RA) narrated: "When the Messenger of Allah would prostrate (for recitation of) the Qur'an, he would say: (Sajada wajhiya lilladhi khalaqahu wa shaqqa sam'ahu wa basarahu bihawlihi wa quwwatihi.) (I have prostrated my face to the One Who created it, and made its hearing and vision, though His ability and power.)" (Grade: Da'if)
حَدَّثَنَا مُحَمَّدُ بْنُ بَشَّارٍ، حَدَّثَنَا عَبْدُ الْوَهَّابِ الثَّقَفِيُّ، حَدَّثَنَا خَالِدٌ الْحَذَّاءُ، عَنْ أَبِي الْعَالِيَةِ، عَنْ عَائِشَةَ، قَالَتْ كَانَ رَسُولُ اللَّهِ صلى الله عليه وسلم يَقُولُ فِي سُجُودِ الْقُرْآنِ بِاللَّيْلِ " سَجَدَ وَجْهِيَ لِلَّذِي خَلَقَهُ وَشَقَّ سَمْعَهُ وَبَصَرَهُ بِحَوْلِهِ وَقُوَّتِهِ " . قَالَ أَبُو عِيسَى هَذَا حَدِيثٌ حَسَنٌ صَحِيحٌ .
Reference: Jami` at-Tirmidhi 580; Book 6, Hadith 37; Vol. 2, Book 1, Hadith 580
2. in Salah
Immediately on completing the Ayaah al-Tilaawah, say Allahu Akbar and go straight into Sajdah, pray Subhaana Rabbi'al A’laa’ (at least) 3 times and while saying Allahu Akbar, immediately stand back up in Qiyaam and carry on with the rest of your Salaah.
It is better to recite some verses as going from Qiyaam to Sajdah to Qiyaam and then immediately into Rukoo' resembles mixing up the order in which Salaah is prayed. However, doing so would not invalidate the Salaah.
Forgot to Perform the Sajdah al-Tilaawah?
If you prayed an ayat of Sajdah in Salah, but have forgotten to perform Sajdah then as long as you are in the "Hurmah (sanctity) of Salah" then you must perform it and then perform Sajdah al-Sahw.
Prostration of Forgetfulness (Sajdat Al-Sahw)- is an extra two prostrations (sajdah) to make up for forgetfulness or a mistake, which is "sahw" in Arabic. While praying, if one makes a mistake that does not nullify the prayer by forgetting something or experiencing confusion about something, one has to perform a Sajdat al-Sahw. A person continues to be in "Hurmah of Salah" until one performs an act that invalidates the Salah (for example, if the Wudo' is broken, by eating or drinking, talking to someone) - in this instance, a person will continue to be in the "Hurmah of Salah" even if one performed Salaam.Rest Allah knows best.
Source: Quran, Ahadith, Learn Islam, Ibn-e-kathir, IslamQA.