There are some prayers in which the person praying on their own, or the person leading others in prayer is recommended to recite the Qur'an alound, and there are other prayers in which it should be recited quietly. As for the person who is following another in prayer, their recitation is always done quietly.
The prayers in which the Qur'an is recited aloud are the first two rak'ah (units) of Fajr(the dawn prayer), Maghrib (the sunset prayer), and'Ishaa' (the evening prayer), as well as Jumu'ah (the Friday congregational prayer), the 'Eid prayer, the prayer for rain, and the prayer at the time of an eclipse of the sun or moon. It is also common to recite voluntary night prayers aloud.
The prayers in which the Qur'an is recited quietly are Dhuhr (the noon prayer), and 'Aṣr (the mid-afternoon prayer), and the voluntary prayers, other than the night prayer. It is also recited quitely in the third and fourth rak'ah of Maghrib (the sunset prayer), and 'Ishaa' (the evening prayer).
Abu Hurayrah (may Allah be pleased with him) narrated:
"[The Qur'an] is recited in every prayer. In those prayers in which Allah's messenger recited aloud for us, we recite aloud for you; and the prayers in which he recited quietly, we recite quietly." [al-Bukhārī: 772]
Some of the many proofs for this from the Sunnah are such as the following:
"The Messenger of Allah (may the peace and blessings of Allah be upon him) used recite al-Fātiḥah and another chapter in the [first] two rak'ah of Dhuhr and ‘Aṣr, and he used to make us hear him [recite] verses sometimes." [al-Bukhārī: 762]
From the previous narration, one can also see that it is permissible for the person leading others in prayer to occasionally make a verse of the Qur'an audible in the prayers where recitation is to be done silently.
When one of the companions of the Prophet (may the peace and blessings of Allah be upon him) was asked:
"Did the messenger of Allah (may the peace and blessings of Allah be upon him) read [verses from the Qur'an] during Dhuhr and 'Aṣr [prayers]? He said: ‘Yes’ They asked: ’How did you know?’ He said: 'With the movement of his beard.'" [al-Bukhārī: 746]
Ibn ‘Abbās (may Alaah be pleased with him) narrated that that Umm Al-Faḍl had heard him recite Sūrah al-Mursalāt, and she said to him:
"My son! Indeed, you have brought memory back to me when you recited this Sūrah! It was the last thing I heard the ؤessenger of Allah (may the peace and blessings of Allah be upon him) recite during Maghrib [prayer]." [al-Bukhārī: 763; Muslim: 462]
al-Barā' ibn 'Aazib (may Allah be pleased with him) said:
"I heard the messenger of Allah (may the peace and blessings of Allah be upon him) recite wat-Teen waz-Zaytoon during ‘Ishā [prayer]...” [al-Bukhārī: 769]
You can also recite loudly (if you wish to) in the tahajjud night prayer (and its variants, like taraaweeh). From the evidences for doing so is the following narration:
"The Prophet (may the peace and blessings of Allah be upon him) came out at night to find Abu Bakr (may Allah be pleased with him) praying in a low voice, and he passed by 'Umar ibn Al-Khaṭṭāb (may Allah be pleased with him) who was praying in a loud voice. Later, when they gathered around the Prophet (may the peace and blessings of Allah be upon him), he said: ‘O Abu Bakr, I passed by you and you were praying in a low voice!’ He said: ‘The One with whom I was conversing hears me, O Messenger of Allah!’ He said to 'Umar: ‘I passed by you while you were praying raising your voice! So he said: "O Messenger of Allah, I wake up the slumbering and repel the devil away.’ The Prophet (may the peace and blessings of Allah be upon him) said: ‘O Abu Bakr, raise your voice a little bit and to 'Umar: lower your voice a little bit." [Abū Dāwood: 1329]