What kind of head covering does a Muslim have to wear? Must he cover all of his hair or only the front of his head? Is wearing a cap in daily life mandatory or Sunnah?

Praise be to Allah, the Lord of the Worlds; and blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Uncovering the head in Salat for men is lawful. So whoever prays without any cover on his head, his prayer is valid and there is no disagreement of scholars over this issue. But some men, who do not usually cover their heads cover them when they pray believing that this is rewardable, but this belief is not supported by evidence. In fact, there is no difference in reward for praying with the head covered or uncovered.
Hadeeth that are reported from the Prophet about the favor of praying with the Amama (turban) over praying without it are weak and therefore cannot be taken as proof in this matter. So, whoever wants to imitate the Prophet in everything - and that is undoubtedly more favorable - he could take the Amama as a part of his normal clothing and wear it in all situations. But wearing the Amama is a habit of tradition and not a Sunnah of worship.
So, if the person does not usually wear a cover over his head, he is not asked to do so when praying; what he should do is wear his best clothes.
As for the cap or hat, it is lawful to put it on the head whether or not it covers the whole head or only part of it, provided it does not resemble the Jewish cap or hat because it is forbidden, as symbols of a non-believer's religion should not be respected or carried out by Muslims.
Allah knows best.