Short answer
A hafiz (female: hafiza) is a Muslim who has memorised the whole Quran by heart, all 114 surahs, and can recite it from memory. The title is an honour and a responsibility: a hafiz must keep revising so as not to forget it.
Memorising the whole Quran is fard kifayah (a communal duty) on the Muslims, so there must always be people who carry it in their hearts. Letting it slip from memory through neglect is blameworthy, and many Hanafi scholars count it a serious sin.
The four schools agree on this.
Explanation
The Arabic word hafiz means guardian or keeper. In Muslim usage it means someone who has memorised the entire Quran. In South Asia the title Hafiz is often used before a name, and in the Arab world the word hamil al-Quran (carrier of the Quran) is also common. Some people memorise it as children over two to four years; others do it as adults.
Why it matters: Allah promised to preserve the Quran (15:9), and one way He has done so is through the millions of people who carry it word for word. The Prophet ﷺ said that the companion of the Quran will be told on the Day of Judgement to recite and rise, and that his rank will be at the last ayah he recites.
The responsibility: the Prophet ﷺ warned that the Quran escapes the memory faster than a tied camel runs off, and told people to keep revising it. A hafiz usually recites a set amount daily and leads or listens to taraweeh in Ramadan to keep it fresh. He also disliked the phrase "I forgot such and such an ayah" and preferred "I was made to forget it".
Being a hafiz is not the same as being a scholar: a hafiz has memorised the text, while a scholar has studied its meaning and law. Both are honoured. To start, see how to memorise the Quran and the hifz planner.
The evidence
Quran, Surah Al-Hijr 15:9
إِنَّا نَحْنُ نَزَّلْنَا ٱلذِّكْرَ وَإِنَّا لَهُۥ لَحَٰفِظُونَ ﴿٩﴾
Indeed, it is We who sent down the Qur'an and indeed, We will be its guardian.
Allah has promised to preserve the Quran, and its memorisers are part of that preservation.
Jami' at-Tirmidhi 2914
Grading to be checked... يقال لصاحب القرآن اقرأ وارتق ورتل كما كنت ترتل في الدنيا ...
"It will be said to the companion of the Quran: Recite and rise, and recite slowly as you used to recite in the world, for your rank is at the last ayah you recite."
The honour of the one who carries the Quran.
Sahih al-Bukhari 5032
Sahih... بئس ما لأحدهم أن يقول نسيت آية كيت وكيت بل نسي ...
"How bad it is for one of them to say, I forgot such and such an ayah. Rather, he was made to forget."
Shows the concern for not losing what one has memorised.
Where the ruling is recorded
- al-Durr al-Mukhtar, Kitab al-Salah, section on recitation. Memorising the amount needed for salah is fard on each person, al-Fatiha and a surah wajib, and the whole Quran fard kifayah.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. 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.