Short answer
Tahajjud is a voluntary (nafl) prayer that is also a sunnah of the Prophet ﷺ, so both words are right. It is not obligatory and you are not sinful for leaving it, but it is the best prayer after the fard. The four schools agree on this.
Tahajjud is a nafl prayer with the status of a recommended sunnah (mustahabb or sunnah ghayr mu'akkadah). It is best prayed after waking from some sleep in the last part of the night, in units of two rakats, and it is not wajib like witr.
The four schools agree on this.
Explanation
All four schools agree that tahajjud (the night prayer) is voluntary. People get confused because two labels are used. Nafl means extra worship beyond the obligatory. Sunnah means the practice of the Prophet ﷺ. Tahajjud is both: it is extra, and he prayed it regularly. It is not a sunnah mu'akkadah (confirmed sunnah) in the way the two rakats before Fajr are, so leaving it is not blameworthy, but it carries great reward.
Can you pray it without sleeping? The word tahajjud refers to rising from sleep, so the ideal is to sleep after Isha and wake in the last third of the night. If you cannot wake, many Hanafi scholars say that nafl prayed after Isha and before you sleep still earns the reward of night prayer (qiyam al-layl), though the waking version is better.
- Rakats: any even number, commonly 2 to 12, prayed in pairs. There is no fixed minimum beyond two.
- Witr: if you trust yourself to wake, keep witr for the end. If you prayed witr before sleeping, you may still pray tahajjud later, but do not repeat the witr.
- Recitation: any surahs you know; longer recitation is better if you can manage it.
For the full method and the best times, read how to pray tahajjud, check tahajjud time tonight, and see the dua for tahajjud.
The evidence
Quran, Surah Al-Israa 17:79
وَمِنَ ٱلَّيْلِ فَتَهَجَّدْ بِهِۦ نَافِلَةًۭ لَّكَ عَسَىٰٓ أَن يَبْعَثَكَ رَبُّكَ مَقَامًۭا مَّحْمُودًۭا ﴿٧٩﴾
And from [part of] the night, pray with it as additional [worship] for you; it is expected that your Lord will resurrect you to a praised station.
The night prayer is described as additional (nafilah) worship.
Sahih Muslim 1163
Sahih... وأفضل الصلاة بعد الفريضة صلاة الليل ...
"The best prayer after the obligatory prayer is the night prayer."
Tahajjud is the most virtuous voluntary prayer.
Sunan Abi Dawud 1439
Graded sahih... لا وتران في ليلة ...
"There are no two witrs in one night."
If you prayed witr early you pray tahajjud after it without repeating witr.
Where the ruling is recorded
- Radd al-Muhtar (Ibn Abidin), Chapter on witr and nawafil, discussion of the night prayer. Night prayer is recommended, and tahajjud is the nafl prayed after sleeping.
- Nur al-Idah with Maraqi al-Falah, Chapter on nawafil. Night prayer is among the recommended prayers, best in the last part of the night.
This answer is pending review by the scholars on our team. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.