Why are some verses missing in the NIV? I'm wondering why some passages in the NIV are missing when compared to the KJV, like:

Matthew 17:21
Matthew 18:11
Mark 9:44
Mark 11:26
Mark 15:28
Luke 17:36

…and more. Also there are some passages where the NIV deletes some phrases as in Matthew 5:44:

NIV: But I tell you, love your enemies and pray for those who persecute you
KJV: But I say unto you, Love your enemies, bless them that curse you, do
good to them that hate you, and pray for them which despitefully use
you, and persecute you

I don't understand the need to change the Bible from what it was for so many years; I personally find the old versions really rich in culture and those express much more than the new versions do. Also in the old versions, the points and commas were really important to not change the original meaning. And the Bible teaches that God is the one that gives us revelation through the Holy Spirit to understand the scriptures, so there's no need to change the Bible to make it 'more understandable'
The NIV has some "missing" verses because it is based on different manuscripts than, say, the King James version.

For some background, we don't have manuscripts of the original writings of the New Testament.  We have copies of them, and as with text that is copied manually, there is room for copyist error, or for people to add or remove things - purposely or inadvertently.  Note that copyist errors don't affect the doctrine of inspiration, inerrancy, or infallibility of the Word.  And even though there are slight changes, it's still reliable for doctrine.

The verses that the NIV "removes" are simply not in the manuscripts used for translating it.

There's more detail available all over the web.  This is one such page.