The verse is 6:4 and is "The Nephilim[a] were on the earth in those days, and also afterward, when the sons of God came in to the daughters of man and they bore children to them. These were the mighty men who were of old, the men of reknown."
Wikipedia notes several translations including "the fallen ones" and "giants." Did other cultures note such a group or is this uniquely from the Bible?
The references are not only in the Bible; they appear in the apocryphal Book of Enoch as well. Several scholars suspect that they have their origins in a pre-Judaic Mesopotamian religion, but I have never seen anything more concrete than that in writing.
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On the translation question, the Hebrew verb niphal means to fall, hence the translation 'fallen ones'. The Septuagint has 'giants'; these mysterious Nephilim are also mentioned in Numbers 13:33
/u/telkanuru has already mentioned their appearance in later literature such as the Book of Enoch. As far as I can tell, there is no comparable reference to Nephilim in comparable Ancient Near Eastern literature of the time. Although A.D. Kilmer has made an argument they should be identified with the Mesopotamian apkallu, who were semi-divine sages. This view comes in "The Mesopotamian Counterparts of the Biblical Nepilim" Perspectives on language & text p 39-43. Winona Lake, Ind : Eisenbrauns, 1987