I often see that when people discuss fearing Hashem they refer to it as "Yiras Shamayim." Why is it called "Yiras Shamayim" and not "Yiras Hashem"?
Partly a comment on DoubleAA's answer, and partly an independent answer:
It is true that in general we find Chazal avoiding the use of Hashem's name where possible, and using alternative terms instead like (Hebrew) "shamayim" or "hamakom," or (Aramaic) "rachamana." We also don't find them using any of these terms as commonly as we use "Hashem" - for example, the expression "brich Rachamana" (corresponding to our "baruch Hashem") appears only eight times in the Gemara, and I've found only one example that might correspond to our "im yirtzeh Hashem" - in Sanhedrin 98a (אם ירצה אדון הזה, see Rashi there, although I think other commentaries explain it differently).
[I once saw in one of R. Avigdor Miller zt"l's books (don't recall which one) that this reflected the lower level of closeness to Hashem that they enjoyed as compared to the era of Tanach (when we find people using His name constantly, attributing events to Him, greeting each other with "May G-d bless you," etc.). In this connection he cites the statement in the Gemara, Yoma 69b: "Since they knew that Hashem is truthful, they wouldn't act falsely towards Him" - and explains that therefore the sages of the era of the second Beis Hamikdash and later wouldn't use terminology that would falsely imply a clear perception of His presence.]
On the other hand, at some later point in Jewish history it became more common to use the term "Hashem" to refer to Him, and to do so more often. (The importance of using phrases such as "baruch Hashem" got a particular boost through the efforts of the Baal Shem Tov, though of course the expression long predates him.) So it may simply be that the phrase "yiras shamayim" is common enough in the Talmudim and Midrashim that it stayed "fossilized" in that form, whereas as it became more common to bless Hashem for events in the course of casual conversation, people naturally would use the more familiar term.
My Rebbe explained that fear of Hashem is not terror of His power over us, but rather awe at how far above us He is. We express that distance by referring to Shamayim which is literally and figuratively unreachable. Thus, Yiras Shamayim is awe and admiration of Hashem's unknowable greatness.
The phrase yiras shamayim shows up many, many times in Talmud era writings, even in conjunction with verses from the Bible that specifically use the phrase yiras Hashem (see Berachot 8a).
My best guess is that it is used to avoid saying God's name. We choose the euphemism of "Shamayim" as an allusion to the verse in Yonah 1:9
ויאמר אליהם עברי אנכי ואת יקוק אלהי השמים אני ירא אשר עשה את הים ואת היבשה:
And [Jonah] said to [the sailors] "I am a Hebrew, and it is the Lord, God of the Heavens [Shamayim], the maker of the land and the oceans whom I fear." (my translation)
In Divrei HaRav (p. 157-158), Rabbi Joseph B. Soloveitchik is quoted as explaining that the concept of fearing God is based on the vast difference between our finite existence and God's infinite existance, whereas the concept of love of God is based on the close personal connection God forges with each of us. Accordingly, he explains, for the latter concept we use the term אהבת השם to signify our being on a "first name basis" with God, while for the former concept we use the term יראת שמים to signify the great distance between us and God.
Yirat Shemayim is not only awe of Hashem, but all of his heavenly creations as well. We look "up" when we feel that awe, towards the heavens in general.
The Kli Yakar Breishis 1:6 mentions the Gemara in Chagiga 12a how when Hashem created the heavens they kept on spreading until Hashem screamed at them and then it stopped spreading. Based on this Kli Yakar I heard from Rabbi Menachem M Lerner of Anshei Sefard in Lakewood that the reason we say Yiras Shamayim is to remember the Yira the Shamayim had and continues to have to this very day, and that is how our Yirah should be from Hashem.