Gensis 6 tells us this of the days before Noah's Flood: Now it came to pass, when men began to multiply on the face of the earth, and daughters were born to them, that the sons of God saw the daughters of men, that they were beautiful; and they took wives for themselves of all whom they chose ... There were giants on the earth in those days, and also afterward, when the sons of God came in to the daughters of men and they bore children to them. Those were the mighty men who were of old, men of renown" (verses 1-2, 4).
Some have taught that this refers to fallen angels interbreeding with human women to produce half-demon giants. But there's a more rational explanation.
Halley's Bible Handbook, a popular study resource, states that "the 'sons of God' (v. 2) are thought to have been either fallen angels ... or leaders in Sethite families [those descended from Adam's son Seth) who intermarried with godless descendants of [Adam's son] Cain" (25th ed., 2000, p. 96).
The first possibility offered here is not really a possibility at all, even though angels are referred to as "sons of God" in Job 38:7 because God is their "Father" through creation. Angels are spirit beings (Hebrews 1 :7), not fleshly creatures. They neither marry nor sexually reproduce (Luke 20:34-36). Also, this explanation would violate the principle made clear in Genesis 1 that each living kind reproduces only "according to its kind." Furthermore, the risen Jesus explained that fallen angels, or demons, are not able to manifest themselves materially like He and the righteous angels can (Luke 24:39; compare verses 40-43; Genesis 18:1-8,16; 19:1). Rather, we see demons in Scripture only possessing individuals or appearing as ghostly apparitions.
The second explanation in Halley's is far more reasonable. Genesis 4 gives the story of Cain and Abel and follows with the genealogical descent from Cain. Genesis 5 is called "the book of the genealogy of Adam" (verse 1). It starts with God's creation of Adam and how Adam's line continued through Seth. As with the angels, Adam was a "son of God" by creation (compare Luke 3:38) — though even more so since Adam was made in God's image (Genesis 1:26; 5:1-3). Of this family line through Seth it is stated, "Then men began to call on the name of the LORD" (Genesis 4:26)—which could also be rendered " ... began to be called after the name of the LORD." Then, in the next chapter, Genesis 6, we see "the sons of God" (men of Seth's godly line in this explanation) intermarrying with "the daughters of men" (women of Cain's ungodly line).
There is also another sensible way to understand what Genesis 6 is describing, in which the phrase rendered "sons of God" should actually be translated "sons of the gods," as the Hebrew word elohim here, plural in form, can sometimes refer to false gods instead of the true God. In this explanation, wicked men referred to as sons of the gods (either pagan worshippers or perhaps men purported to be demigods, as ancient rulers often claimed of themselves) forcibly "took" innocent women as wives–an example perhaps of the evil conduct of the day.
In any event, human beings were clearly the problem here–not angels. God says amid the verses of Genesis 6 quoted above, "My Spirit shall not strive with man forever" (verse 3) and "I will destroy man whom I have created from the face of the earth" (verse 7). Thus, the "giants" mentioned must have been human also—descendants of Adam and Eve (compare Acts 17:26).
It's possible that the word translated "giants" here, nephilim–from a root meaning to fall or make to fall–could refer to tyrants. But the word is used of people of giant stature later in Numbers 13:33. So the very powerful, and possibly very tall, people of Genesis 6 were destroyed in the Flood. But there would be other giants following the Flood, who were descended, just as everyone else in the post-Flood world, from Noah—again, not angels (compare Deuteronomy 2:20-21; 3:11). Consider Goliath, whom David slew. He was more than nine feet tall (1 Samuel 17:4). But he was still just a man (verses 24-25, 33)–not some human-demonic hybrid.