I decided to build a canonical answer to this question, since it seemed that all three answers had something to offer.
Greek and Hebrew
The Hebrew word yare (Strongs H3372) carries a number of meanings. There is both the definition being "terrified" or "afraid" along with the definition of having "reverence", "awe", or "respect".
In Greek, the word phobeō (Strongs G5399) carries similar meanings of both "being afraid" and "having reverence or awe".
So, per the definitions of the words in their original languages, we can see that we have to take the word in context to determine whether this is the sense of "fear" or "reverence" that should be understood.
When, for example, Deuteronomy 6:13 says that we should
fear the Lord your God, this clearly is referring to having "awe, reverence, and respect". By comparison, when in Genesis 26:24 God says
do not fear, for I am with you, he's clearly saying that we shouldn't be terrified.
However, often these meanings can intertwine a bit:
Psalms 2:11 (ESV)
Serve the LORD with fear, and rejoice with trembling.
In this passage, there's both the awe and respect along with a sense of being afraid.
To further support this idea, Merriam Webster shows fear as containing both of the definitions mentioned above:
1 archaic : frighten
2 archaic : to feel fear in (oneself)
3 : to have a reverential awe of (fear God)
4 : to be afraid of : expect with alarm (fear the worst)
So, "fear" carries both the definition of being afraid (#4) and showing awe and respect (#3)
In fact, as a verb, "fear" is more about awe and respect; as a noun it's more commonly used as
an unpleasant often strong emotion caused by anticipation or awareness of danger
"Fear" is not "dread"
It should also be noted that there is a difference between fear (as in "fear and trembling") and dread.
Judaism, continued Rabbi Soloveichik, requires yirat shamayim, but not fear. The Bible never commands us to have pahad for God, but only yirah; and the Talmud inculcates the virtue of yirat shamayim, not pahad shamayim. While Judaism does not advocate pahad, "the whole Torah in its entirety is founded on the foundation of yirah."
Yirat Shamayim: The Awe, Reverence, and Fear of God Edited by Marc Stern; p. xxi (Introduction)
In this quote, yirah is translated fear while pahad is more accurately translated as "dread". The phrase yirah shamayim is "fear of heaven" (compared to pahad shamayim, the "dread of heaven").
This simply shows that while there may be a component of being afraid in the word "fear", there's not that complete sense of dread. (Otherwise pahad would have been used.)
"Awe and reverence" generally isn't used for "fear" outside of religious circles. In the popular media (in particular), "fear" is generally used as a lesser version of "dread" and does not take on the connotation of "reverence". It's this common, popular use of "fear" as "lesser dread" that causes the confusion about this word.
"Fear" carries two meanings: a lesser form of dread, and an awe and respect. It is often used both ways in the bible. Sometimes it carries both meanings. Since it's often impossible to separate the difference in context, it's always translated as "fear", since the English version of the word also carries both definitions (regardless of the fact that "awe and respect" is much less common than "dread").