ANSWER: While the word Nephilim is not found in the KJV or most other English Bible translations, it does appear in the original Hebrew of the Old Testament. It appears once in the book of Genesis and once in the book of Numbers.
There were giants (nephilim) in the earth in those days; and also after that, when the sons of God came in unto the daughters of men, . . .
And there we saw the giants (nephilim), the sons of Anak, which come of the giants (nephilim): and we were in our own sight as grasshoppers, and so we were in their sight (Genesis 6:4, Numbers 13:33, KJV)
The Hebrew word nephilim, Strong's Concordance #H5303, is defined as someone who is a feller, meaning a person who is a tyrant or bully. The Brown-Driver-Briggs Hebrew Definitions reference simply translates the word as "giants."
Giants did indeed exist before and after Noah's flood. These large people were especially abundant when the earth was experiencing a population explosion (Genesis 6:1) prior to the deluge. Conservative estimates place the world's population, before it was covered with water, at around one billion!
One common question is who were the "sons of God" who married women and produced the nephilim. Some believe they were demons who took the form of human flesh and produced giants through human females. This theory does not make any sense, however, since there is no place in Scripture where God calls demons his "sons."
Additionally, there is a giant (pardon the pun) problem with the belief that angels and women were able to produce a race of supersized humans called Nephilim. Scripture says the sons of God took wives for themselves, meaning they were married, and together they produced such children (Genesis 6:2). It does NOT say these women were raped by angels and then produced bastard giants.
Marriage is a Holy, binding relationship between a man and a woman God instituted in Eden (Genesis 2:24 - 25). There is not even a HINT in scripture that He ever spoke about or approved of angelic beings somehow marrying humans and then allowing them to creat the Nephilim.
The possibility of angels and humans marrying and producing nephilim was also indirectly addressed by Jesus. He stated that marriage was a human institution that will not exist after people are resurrected and changed into spirit like the angels (Matthew 22:30). Angelic beings are incapable of marriage and cannot reproduce through procreation.
The Bible informs us, in places other than Genesis, that God's sons are those who have his Holy spirit dwelling in them (John 1:12, Romans 8:14, 1John 3:2). The nephilim or giants were produced by righteous men who foolishly married worldly women (see Genesis 6:2). Instead of relying on God to guide their marital decisions, these men picked mates they lusted after. They became, as the apostle Paul states, "unequally yoked" with unbelieving women (1Corinthians 7).