For the Persian lyric poet (Khwajeh Shams al-Din Muhammad Hafez-e Shirazi), see Hafez.
In Islamic culture, a hafiz (literally, "guardian" in Arabic), is someone who has memorized the entire Quran. The plural of hafiz is huffaz, and the feminine form is hafiza.
The prophet Muhammad lived in a society of nomadic tribes at a time when Arabic had not yet developed a full written language. Thus, when he proclaimed the verses later collected into the Quran, it was crucial that many people memorize them, in order for them to be preserved and passed on. It is for this reason that the hadiths (non-Quranic sayings and deeds of the prophet) suggest that a person who memorizes the entire Quran will receive a special place in paradise, as will their parents, and that they will also be able to intercede on behalf of 10 other people.
Even today, when the Quran has long-since been written down, huffaz are accorded great respect and remain an important part of Islamic society and ritual, and memorizing the Quran remains at the core of the curriculum in many Muslim madrassa schools. As a result, despite the fact that the Quran is about 80,000 words long, there are many, many thousands of huffaz in the world today.
Finally, it is worth noting that the word "hafiz" was not always used in its current sense. The original Arabic term for someone who had memorized the entire Quran was hamil ("one who carries"), whereas "hafiz" was used for someone who had memorized hadiths. But over time "hafiz" came to overtake "hamil" as the most common term for someone who had memorized the Quran.