Sin nature: referring to an inability to do what is right
I think sin nature refers to our inability to "Be holy, as God is Holy", brought about through our natural inclination to do what is wrong:
"Jesus said unto him, Thou shalt love the Lord thy God with all thy heart, and with all thy soul, and with all thy mind. This is the first and great commandment. And the second is like unto it, Thou shalt love thy neighbour as thyself. On these two commandments hang all the law and the prophets." (Mat.22: 37-40).
God coammands that we are to "be holy, as God is holy" (Lev.11: 44), and Jesus said we are to "be perfect, as your Father in heaven is perfect" (Mat.5: 48).
The apostle Paul spoke about our inability to do this through human endeavor, and then he gave the solution to our sin nature:
"For I do not do the good that I desire; but the evil which I do not will, that I do." (Rom.7: 19).
This refers to (a) our inability to do what is right; and (b) our natural inclination to do what is wrong. Paul then gives the solution to our problem:
"O wretched man that I am! Who shall deliver me from the body of this death? I thank God through Jesus Christ our Lord!..." (Rom.7: 24-25).
Jesus said we are to remain in and abide in Him because just as a branch that does not remain in the vine cannot produce good fruit, without Jesus we cannot produce good fruit (Joh.15: 4). This is because the sin nature in us won't permit us to bring forth good fruit - because our sin nature will continually cause us to produce bad fruit.
Before the days of the New Covenant of grace, God's people were told to sanctify themselves (make themselves holy - Lev.11: 44) - and this through their own human endeavor, but it was an inpossible thing for them to do, because of their sin nature. This is why the apostle Paul said that the Law of God was our schoolmaster leading us to Christ, for in Christ we are NOT sanctified by our own attempts to sanctify ourselves - we are sanctified (made holy) by the Holy Spirit of Jesus through our faith in Jesus (Rom.11: 56; 1Cor.1: 2; 1Cor.6: 11, etc).
Andrew