Gemara

Rav Kahana took a turban in place of the five Sela'im for Pidyon ha'Ben (redemption of a firstborn son). To him, it was worth five Sela'im.

(Rav Ashi): Only a great man like Rav Kahana, who needs a turban, may do so.

29b: Rav Chisda praised Rav Hamnuna in front of Rav Huna. Rav Huna saw that Rav Hamnuna did not wear a turban like the custom of married men. He refused to look at Rav Hamnuna until he married.

Shabbos 118b (Rav Huna brei d'Rav Yehoshua): I will receive (reward) for not going four Amos with my head exposed.

138b (Rav Sheshes brei d'Rav Idi): One may wear Si'ana (a kind of hat) on Shabbos.

Question: It was taught that it is forbidden!

Answer (#1): It is forbidden (due to making an Ohel) only if it spreads out a Tefach from his head.

Objection: Is one liable for spreading a Talis a Tefach over his head?! (Rosh's text - is this forbidden?!)

Answer: A tight hat is different.

Rishonim

Rambam (Hilchos Tefilah 5:5): One may not stand in prayer when his head is exposed. It is the practice of Chachamim and their Talmidim to pray only when they are covered.

Rosh (Shabbos 20:3): Si'ana is a hat to shade the head. Rashi explains that we conclude that it is forbidden if it does not fit well. We are concerned lest wind blow it off his head and he will carry it four Amos. R. Tam's text does not say 'Ela'. The Gemara answered that we are concerned for an Ohel only when it is tight and does not bend. Also R. Chananel explains this way.

Poskim

Shulchan Aruch (OC 8:2): It is proper to cover one's head with the Talis.

Beis Yosef (DH u'Mechaseh): The Tur says (in the name of Ba'al ha'Itur) 'in order that his head will not be exposed.' He does not mean totally exposed, for this Halachah is unrelated to Tzitzis. Also, one goes to the bathroom before putting on a Talis! Rather, the custom of pious people is to cover their heads with a turban or Talis, like we find in Kidushin (29b). Surely, Bochurim did not go with exposed heads! Rather, only married men covered with a turban or Talis, for this brings to Yir'as Shamayim, like we find in Shabbos (156b). Perhaps the Tur suggests covering the head with a Talis of a Mitzvah, for this is the ideal Mitzvah of Tzitzis.

Rebuttal (Gra 6): If the head was covered by a smaller covering, the Tur would not call this 'uncovered.' Throughout Shas we find that people covered their heads with a Sudar. The Rif (Berachos 44b) says that Birkas Tzitzis is after Malbish Arumin, and afterwards one blesses Oter Yisrael b'Sif'arah when putting on a turban. (Note: This is not in our text of the Rif. Our Gemara (Berachos 60b) says that one blesses on the turban before Tzitzis.)

Taz (3): It seems that the Tur says that one should be careful to cover the head (with something else) even when wearing a Talis, for the Talis often slips from the head.

Shulchan Aruch (91:4): A cap woven of straw is considered a covering.

Terumas ha'Deshen (10): The Torah requires a (married) woman to cover her hair. Nevertheless, a basket on her head suffices even for Reshus ha'Rabim, and nothing is needed between Chatzeros. All the more so, a straw cap, which is woven tighter with less gaps than a basket, suffices for Berachos. A woman's hair must be covered lest it arouse men, therefore we are not concerned in a Chatzer, for many do not pass through. Covering for Berachos is due to frivolity towards Shamayim, i.e. saying His name without fear. We do not distinguish between a house and Reshus ha'Rabim.

Ateres Zekenim (2:1): A toupee is a proper covering, but one should try to have a different covering, for it looks like his head is uncovered.

Shulchan Aruch (ibid.): Putting one's hand on his head is not a covering.

Terumas ha'Deshen (ibid.): If Hash-m's name was written on one's body, he may not be naked in front of it. Covering it with his hand would suffice, but we decreed lest he forget and remove his hand. The Or Zaru'a says that a hand on the head does not permit Berachos. When R. Tam wanted to bless on a drink while bathing, he used a garment to separate between his heart and his Ervah. He did not rely on his hand! The hand cannot cover the body because it is part of the same body.

Rebuttal (Maharshal, ibid.): A hand does not separate the heart from the Ervah, for people often put their hand over their heart. Also, it might look like he did not cover his heart exactly. Putting one's hand on his head is recognizable. Since it is only Midas Chasidus, one may be lenient, especially if he does so only occasionally.

Support (Taz 8:3): Putting the hand on the head shows that one may not sit with his head uncovered. However, this does not suffice for saying His name.

Magen Avraham (4): One should not rely on the Maharshal.

Shulchan Aruch (ibid.): It suffices for Levi to put his hand on David's head.

Beis Yosef DH v'Chasuv bi'Terumas): The Terumas ha'Deshen (10) permits a cap, but not one's hand, for it is from the same body, and the body cannot cover itself. This implies that another's hand helps.

Igros Moshe (OC 1:1): Ha'Elef Lecha Shlomo requires covering the majority of the head to walk four Amos. The hand suffices even for Berachos, even though it covers only a minority!

Rebuttal (Doleh u'Mashkeh p.2): Also the Chazon Ish requires the majority. We must say that the discussion is about covering the head with both hands.

Shulchan Aruch (5): One may not stand in prayer when his head is exposed.

Mishnah Berurah (12): Nowadays, one must wear a hat during prayer. A Kipah does not suffice, for people do not stand in front of important people with only a Kipah.

Shulchan Aruch (301:41): Some forbid to go out (in Reshus ha'Rabim) on Shabbos with a hat to protect from the sun, unless it is tight fitting, goes deep on his head or is tied under his neck with a strap. If not, we are concerned lest wind blow it off and he will carry it four Amos.

Beis Yosef (DH Kova): A hat used to cover his head is permitted, for one does not walk with his head uncovered

Mishnah Berurah (153,154): It is forbidden if he has a cap underneath. The same applies to our hats. The Taz is stringent outside an Eruv, even if it is not Reshus ha'Rabim.

Kaf ha'Chayim (241): Eliyahu Rabah permits even with a cap underneath if the practice is not to walk outside without a proper hat.