What is the actual temperature of Yad Soledes?
As with everything, there are many different opinions. This is probably in large part because the Talmud (Shabbat 40A) doesn't give a temperature, but rather says that Yad Soledet Bo is "hot enough to scald a baby's stomach".
Here are some of the temperatures given:
From here, based on Shabbos KeHalachah by Rabbi Y. Farkash:
Most Poskim agree that Yad Soledes Bo is between 40 and 45 degrees Celsius (104-113 Fahrenheit), or approximately 110 degrees Fahrenheit. (Shabbos KeHalachah, p. 24.)
There are (at least) 15 opinions from various Poskim brought here, each one with sources (which I didn't look up). I'll quote the ones that have numbers:
Chelkas Yakov (YD 44:3): If water gets between 50 and 70 degrees (Celsius), this is Yad Soledes Bo -- (between 122 and 158 Fahrenheit).
(OC 4:74): It is proper to be stringent to be concerned lest 43 degrees is Yad Soledes Bo. There is a Safek until 71 degrees. That is surely Yad Soledes Bo, even to be lenient. -- (43C == 109F and 71C == 160F)
(YD 2:52): Yad Soledes Bo is about 79 degrees, to be stringent. (79C == 174F)
(YD 3:31): Surely, 74 degrees is Yad Soledes Bo. (74C == 165F)
Minchas Yitzchak (7:61): Sixty degrees is surely Yad Soledes Bo. (60C == 140F)
Minchas Shlomo: I say that there is no concern less than 45 degrees. (45C == 113F)
"Yad soledes bo" is the lowest temperature at which halacha considers cooking to take place. It's defined as the temperature that would make your hand uncomfortable. This can vary a good degree between people (as experiments with hot showers, or drinking hot tea, have shown), so various rabbis have suggested certain standard numbers.
If I recall correctly, anything less than 113F is considered certainly not "yad soledes." Rabbi Shlomo Zalman Auerbach deduced this from the halacha that a duck has a very high internal temperature, yet it's not "yad soledes" (to cause problems from blood getting "cooked" into the knife used to slaughter it); assuming the duck could have a mild fever, that gets us to 113F.
Ask a person on the street what temperature is uncomfortable, and they'll usually tell you 120F; if you have a question between 113 and 120, ask your rabbi.
On the flipside, we don't consider something "definitely yad soledes" for the purposes of leniency (e.g. this was already "cooked" once) unless it's a much higher temperature; I seem to recall 160F or maybe 170F at the very highest for that purpose.