The prohibition of bathing/showering on Shabbos in hot water:
A. Bathing in hot water heated through fire:
The Sages forbade bathing ones entire body or majority of it with hot water whether [the water] is within a vessel [or] whether it is in the ground, [and] even if it was heated from before Shabbos.
The reason for this prohibition is: because [back then] the people in charge of the bathhouses would heat up [the water] on Shabbos and claim that they heated it before Shabbos.
Now, although the Jewish people are not [to be] suspected to advertently desecrate Shabbos, nevertheless these [people in charge of the bathhouse] would place wood [in the fire that is] under the cold water before Shabbos, close to dark, and the woods would burn throughout the entire Shabbos thus heating up the water that is above it. Now, [placing the wood so close to Shabbos] is forbidden because of a decree that one may come to forget and stoke the coals after Shabbos has begun. Therefore [the Sages] forbade anyone from bathing in hot water even if it is not [being done] in a public bathhouse [which is run by caretakers].
B. Showering in hot water heated through fire:
[Furthermore] even to pour water on one’s body to rinse off [is forbidden] even though this is not the usual way of bathing in a bathhouse in which one enters his body and limbs into the water.
C. Bathing in hot water heated through fire one limb at a time or minority of one’s body:
Washing each limb individually: Even to bathe each limb individually and not majority of his body at once [is forbidden].
The reason for this is: because the Sages did not differentiate within their decree [regarding different forms of washing] since [all the forms] slightly resemble bathing in a bathhouse in which one bathes his entire body or majority of it.
Bathing minority of one’s body: However it is permitted to bathe in hot water [that was permissibly heated from before Shabbos] ones face, hands and feet or other limbs so long as one does not [in total] wash majority of his body. [Regarding washing limbs that contain hair see Halacha 7 Q&A there!] However hot water which was heated on Shabbos even in a permitted way is forbidden to wash in even one limb and even if the [water] was only slightly heated that it is not Yad Soledes.
Summary-Bathing/Showering on Shabbos in hot water
Water that was heated before Shabbos:
Bathing majority of one’s body: Is Rabbinically forbidden even if the water was heated from before Shabbos, and even if one washes each limb individually.
Bathing minority of one’s body: Is permitted to be bathed/washed with water that was permissibly heated from before Shabbos.
In hot water heated on Shabbos:
Even if the water was heated in a permitted way [such as in the sun, or to less than Yad Soledes] it is forbidden to wash even one limb in that water. Thus one may not dry his hands near a source of heat as doing so heats up the water on his hands.
If one placed cold water onto his blech right before Shabbos, and it was thus only heated on Shabbos, may one bathe minority of his limbs in such water?
Yes, as since no action was done to the water on Shabbos it is considered heated from before Shabbos.
However there are Poskim which prohibit washing in such water in this case. Furthermore some say that even if the water was warmed before Shabbos but retained its heat due to its remaining over a flame and the like, then it is considered as if it were heated on Shabbos.
If one mixed hot water that was heated from before Shabbos with cold water [in a way permitted to be done, such as cold pouring water into a Keli Sheiyni], may one bathe minority of his body with this water on Shabbos?
D. Bathing in water that was not heated through fire:
Hot Springs: All the above is with regards to water heated through fire, however the hot water [from the] springs of Tiberius are permitted for one to even dip his entire body inside of. [See Halacha 3 regarding if this is allowed when done for healing purposes!]
Hot water of Tiberius that is in a vessel: However it was only permitted [to bathe] in the hot waters of Tiberius when [the water] is in the ground, however when it is in a vessel then it has the same laws as water heated over fire being that all water that is in a vessel is not recognizable if it was [heated] from a fire or if it is from the hot springs of Tiberius, and there is thus [reason] to decree on these [hot waters of Tiberius] because of the hot waters [heated through fire]. [See Q&A regarding water heated through the sun!]
A roofed spring:
First opinion: [However] there are opinions which say that [the Sages] only permitted [washing oneself in] the hot springs of Tiberius when the area is not roofed, however if the area is roofed then it is forbidden.
Their reason is: because [a roofed spring] causes one to sweat and sweating is forbidden on Shabbos as will be explained later on.
Second opinion: [However] there are opinions which say that it is permitted to sweat in the hot waters of Tiberius.
Their reasoning is: because [the Sages] only forbade one from [actively causing himself to] sweat due to the decree made against bathing, as will be explained later on, and thus since it is permitted to bathe in the hot springs of Tiberius then certainly it is permitted to sweat in them.
The Final Ruling: Regarding the final ruling, in [a controversy over] a Rabbinical prohibition one may follow the lenient opinion.
The custom today by all springs:
Due to that not everyone knows to beware in all the Halachic matters involved in bathing [as brought in 2B], therefore the custom spread in these provinces to not bathe at all on Shabbos, not even with cold water [and certainly not by hot springs of Tiberius] even though there is no prohibition involved from the letter of the law [in doing so].
From the letter of the law one may bathe even his entire body in a natural hot spring, as long as the water is not in a vessel. One may be lenient to bathe in it even if the spring is under a roof. However the custom is to avoid bathing even in cold water.
May one bathe in water that was heated by sun, such as in one’s solar boiler?
Heated from before Shabbos: It is forbidden to bathe majority of one’s body in such water even if it was heated before Shabbos, as since the water is within the boiler it has the same status as spring water that is in a vessel. [See footnote for other opinions] However one may wash minority of one’s limbs in such water if the water became heated by the sun before Shabbos. However one must beware not to transgress the cooking prohibition in doing so, as cold water enters into the boiler upon opening it. It is thus only allowed to be opened if either a) The water in the boiler is below Yad Soledes [110 F.] or b) One shut off the water pipe which adds more cold water to the boiler.
Heated on Shabbos: If the water was heated by the sun on Shabbos then it is implied from Admur that it is forbidden to wash even minority of one’s body with it. However there are Poskim which rule that even when heated on Shabbos it is allowed to bathe in such water minority of one’s body.
E. May one bathe in warm water?
The letter of the law: It is certainly permitted to bathe in warm water heated from before Shabbos as even the waters of a river in the summer are considered warm, or [waters] that are in a [heated] house in the winter [are considered warm, and nevertheless are permitted to wash with on Shabbos]. [See Q&A regarding the definition of warm water.] However hot [or even warm] water which was heated on Shabbos even in a permitted way is forbidden to use to wash even one limb, even if the [water] was only slightly heated that it is not Yad Soledes.
The custom today: Due to that not everyone knows to beware in all the Halachic matters involved in bathing [as brought in 2B], therefore the custom spread in these provinces to not bathe at all on Shabbos, not even with cold water even though there is no prohibition involved from the letter of the law [in doing so].
From the letter of the law it is permitted to bathe in warm water that was heated from before Shabbos, although the custom is to avoid doing so. However water which was heated on Shabbos [whether by fire or by the sun] is forbidden to bathe in even if only warm.
What is defined as warm water as opposed to hot water?
The Tehilah Ledavid learns that even less than Yad Soledes is considered hot, and thus seemingly so long as the water is warmer than body temperate [98.6], or its heat is felt in it, then it is forbidden just like hot water. So is implied also from other Achronim. However there is room to learn from Admur that so long as the water is less than Yad Soledes then it is not considered hot and is thus allowed.
 SSH”K 14/3 in name of Rav SZ”A
Sheivet Halevy 1/58, Betzeil Hachachma 4/136, brought in Piskeiy Teshuvah 226/1
 Sheivet Halevy ibid.
 Ketzos Hashulchan 133 footnote 2
 Minchas Shabbos
 Halacha 1
 Halacha 2
 So rules SSH”K 14/3 in name of RSZ”A based on the stringency of Admur regarding sun heated water.
The Tehila Ledavid [326/3] however rules that water heated through sun is more lenient than spring water that is within a vessel, and is permitted to be fully bathed in on Shabbos.
 So is implied from the next Halacha [E] regarding warm water in which Admur rules in the Mahdurah Basra that if the water was warmed by the sun before Shabbos it is allowed, as proven from the fact that one may bathe in a warm river on Shabbos, thus implying that if the water got heated on Shabbos through the sun then this would not be allowed. So understands also Tehila Ledavid 326/3 and SSH”K 14 footnote 17.
 Tehila Ledavid 326/3, thus arguing on his understanding of Admur. SSH”K 14/3
 As since one did no action to heat the water it is considered as if it were to be heated before Shabbos. [SSH”K 14/3]
 Supplement from Mahadurah Basra chapter 259 Page 884 in the new Shulchan Aruch.
So rules also the Sefer Tehila Ledavid, and Minchas Shabbos, brought in Ketzos Hashulchan 133 footnote 1, and in SS”K chapter 14 footnote 3. This ruling is also found in the Chacham Tzevi 11 which forbids for women to immerse in water on Shabbos that is called hot, rather it must be cold or slightly warm. So rules also Nodah Beyehudah Tenyana Orach Chayim 24 that women may bathe in warm Mikvaos but not hot Mikvaos.
However other Poskim rule that warm water is forbidden to bathe in just like hot water [Rambam brought in Biur Halacha. However the M”B himself  writes that bathing in warm water is allowed [for Mikveh]. Some however write that in truth the M”B holds that only water in which some coldness is felt is allowed, and it is by this warmth that he allows to bathe, however water which its heat is felt is forbidden as writes the Rambam, as he brings in Biur Halacha. [Az Nidbaru, brought in Piskeiy Teshuvos 226/2].
 See Previous Halacha Q&A there!
 Ketzos Hashulchan 133 footnote 1, and SS”K chapter 14 footnote 3
 The Aruch Hashulchan [326/3] writes that although the measurement of hot water is not brought anywhere, it most likely refers to all waters that people call hot. This ruling is also found in the Chacham Tzevi 11 which forbids for women to immerse in water on Shabbos that is called hot, rather it must be cold or slightly warm.
 In Halacha 4 [brought above in Halacha 1C] regarding water heated on Shabbos Admur mentions “even if it will not be heated to Yad Soledes”, implying that before Shabbos only water that is heated to Yad Soledes is forbidden. [Ketzos Hashulchan ibid]