The Nephilim (i.e. from the Hebrew word naphal which means 'to fall') is what the Hebrew word for "giants" is in the KJV.
Gen 6:4
4 There were giants in the earth in those days; and also after that, when the sons of God came in unto the daughters of men, and they bare children to them, the same became mighty men which were of old, men of renown.
KJV
The phrase "men of renown" means 'men of the name, or legend'. What legend?
The 19th century Christian scholar E.W. Bullinger has an interesting Appendix on the Nephilim in The Companion Bible:
"Appendix 25 THE NEPHILIM, OR "GIANTS" OF Gen 6, &c
The progeny of the fallen angels with the daughters of Adam (see notes on Gen 6, and Ap. 23) are called in Gen 6, Ne-phil-im, which means fallen ones (from naphal, to fall). What these beings were can be gathered only from Scripture. They were evidently great in size, as well as great in wickedness. They were superhuman, abnormal beings; and their destruction was necessary for the preservation of the human race, and for the faithfulness of Jehovah's Word (Gen 3:15).
This was why the Flood was brought "upon the world of the ungodly" (2 Peter 2:5) as prophesied by Enoch (Jude 14).
But we read of the Nephilim again in Num 13:33: "there we saw the Nephilim, the sons of Anak, which come of the Nephilim". How, it may be asked, could this be, if they were all destroyed in the Flood? The answer is contained in Gen 6:4, where we read: "There were Nephilim in the earth in those days (i.e. in the days of Noah); and also AFTER THAT, when the sons of God came in unto the daughters of men, and they bare children to them, the same became [the] mighty men (Heb. gibbor, the heroes) which were of old, men of renown" (lit. men of the name, i.e. who got a name and were renowned for their ungodliness).
So that "after that", i.e. after the Flood, there was a second irruption of these fallen angels, evidently smaller in number and more limited in area, for they were for the most part confined to Canaan, and were in fact known as "the nations of Canaan". It was for the destruction of these, that the sword of Israel was necessary, as the Flood had been before.
As to the date of this second irruption, it was evidently soon after it became known that the seed was to come through Abraham; for, when he came out from Haran (Gen 12:6) and entered Canaan, the significant fact is stated: "The Canaanite was then (i.e. already) in the land." And in Gen 14:5 they were already known as "Raphain" and Emim", and had established themselves at Asteroth Karnaim and Shaven Kiriathaim.
In ch. 15:18-21 they are enumerated and named among Canaanite Peoples: "Kenites, and the Kenizzites, and the Kadmonites, and the Hittites, and the Perizzites, and the Rephaims, and the Amorites, and the Girgashites, and the Jebusites" (Gen 15:19-21; cp. Ex 3:8,17; 23:23. Deut 7; 20:17. Josh 12:8).
These were to be cut off, and driven out, and utterly destroyed (Deut 20:17. Josh 3:10). But Israel failed in this (Josh 13:13; 15:63; 16:10; 17:18. Judg 1:19,20,28,29,30-36; 2:1-5; 3:1-7); and we known not how many got away to other countries to escape the general destruction. If this were recognized it would go far to solve many problems connected with Anthropology.
As to their other names, they were called Anakim, from on Anak which came of the Nephilim (Num 13:23), and Rephaim, from Rapha, another notable one among them. From Deut 2:10, they were known by some as Emim, and Horim, and Zamzummim (v. 20, 21) and Avim, &c.
As Rephaim they were well known, and are often mentioned: but, unfortunately, instead of this, their proper name, being preserved, it is variously translated as "dead", "deceased", or "giants". These Rephaim are to have no resurrection. This fact is stated in Isa 26:14 (where the proper name is rendered "deceased," and v. 19, where it is rendered "the dead"). It is rendered "dead" seven times (Job 26:5. Ps 88:10. Prov 2:18; 9:18; 21:16. Isa 14:8; 26:19). It is rendered "deceased" in Isa 26:14.
It is retained as a proper name "Rephaim" ten times (two being in the margin). Gen 14:5; 15:20. Josh 12:15 (marg.). 2 Sam 5:18,22; 23:13.& b31 nbsp; 1 Chron 11:15; 14:9; 20:4 (marg.). Isa 17:5. In all other places it is rendered "giants", Gen 6:4; Num 23:33, where it is Nephilim; and Job 16:14, where it is gibbor (Ap. 14. iv). By reading all these passages the Bible student may know all that can be known about these beings.
It is certain that the second irruption took place before Gen 14, for there the Rephaim were mixed up with the five nations or peoples, which included Sodom and Gomorrha, and were defeated by the four kings under Chedorlaomer. Their principal locality was evidently "Ashtaroth Karnaim"; while the Emim were in the plain of Kiriathaim (Gen 14:5).
Anak was a noted descendant of the Nephilim; and Rapha was another, giving their names respectively to different clans. Anak's father was Arba, the original builder of Hebron (Gen 35:27. Josh 15:13; 21:11); and this Palestine branch of the Anakim was not called Arbahim after him, but Anakim after Anak. They were great, mighty, and tall (Deut 2:10,11,21,22,23; 9:2), evidently inspiring the ten spies with great fear (Num 13:33). Og king of Bashan is described in Deut 3:11."
(from E. W. Bullinger's Companion Bible: Notes and Appendices. Biblesoft Formatted Electronic Database Copyright © 2014 by Biblesoft, Inc. All rights reserved.)