Because it was customary for idol worshippers to bow fully to the ground before their idols, Jews refrain from bowing down (with the exception being during the Yom Kippur service). According to the sages, there are several ways a person might bow:
"Kidah means falling upon the face, as it says (I Kings 1:31), ‘Then Batsheva bowed with her face to the earth." Keri’ah means going down upon the knees, and so it says (I Kings 8:54), ‘[Solomon rose] from kneeling on his knees.’ Hishtachavayah is spreading out the hands and feet, as it says (Genesis 37:10), ‘Shall I and your mother and your brothers come to prostrate ourselves before you to the earth?’" (Megillah 22b).
The Jewish aversion to bowing stems from two Biblical sources. The first is Leviticus 26:1, which states : "...neither shall you place any figured stone in your land, to bow down unto it..." The second traditional source is Mordechai’s refusal to bow down to Haman in the Book of Esther.
In the daily prayer services, however, Jews not only bow four times during the Amidah (silent, standing prayer), but at several other points in the daily service as well (Barchu, Aleinu). It is therefore important to note that the Hebrew term specifically prohibited in Leviticus 26 is Hishtachavayah, full prostration of hands and feet. Nevertheless, Jews tend to refrain from bowing in any other manner as well.
Practically, of course, there are exceptions. One might bow in front of a king or queen if that is the customary way of paying respect to royalty. If one were in Japan, one might bow when greeting others, as is the custom of the country. One might not, however, bow to any animal, in front of a statue or any other inanimate object.