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The way I see it the "soul" is the lifeforce within the being. Even animals have what can be called a lifeforce - though some are significantly more sentient than others (e.g., humans vs. worms). In humans there is a third part (we're a "triune" being in God's image after all), called the "spirit". Those who are unsaved have a dead spirit nature (the flesh) that is a slave to sin but when we come to saving faith in Jesus Christ, our spirit comes alive (we are born again) and the Holy Spirit indwells us:

But because of His great love for us, God, who is rich in mercy, made us alive with Christ, even when we were dead in our trespasses. It is by grace you have been saved! (Ephesians 2:4,5)

And having been buried with Him in baptism, you were raised with Him through your faith in the power of God, who raised Him from the dead. When you were dead in your trespasses and in the uncircumcision of your sinful nature, God made you alive with Christ. He forgave us all our trespasses, having canceled the debt ascribed to us in the decrees that stood against us. He took it away, nailing it to the cross! (Colossians 2:12-14)

While we are still physically alive, we will have contention between the old sin nature and the new spirit nature. It is what Paul taught in Romans chapter 7 and he was already saved for 25 years when he wrote that. He explains:

For we know that the law is spiritual: but I am carnal, sold under sin. For that which I do I allow not: for what I would, that do I not; but what I hate, that do I. If then I do that which I would not, I consent unto the law that it is good. Now then it is no more I that do it, but sin that dwelleth in me. For I know that in me (that is, in my flesh,) dwelleth no good thing: for to will is present with me; but how to perform that which is good I find not. For the good that I would I do not: but the evil which I would not, that I do. Now if I do that I would not, it is no more I that do it, but sin that dwelleth in me.

I find then a law, that, when I would do good, evil is present with me. For I delight in the law of God after the inward man: But I see another law in my members, warring against the law of my mind, and bringing me into captivity to the law of sin which is in my members. O wretched man that I am! who shall deliver me from the body of this death? I thank God through Jesus Christ our Lord. So then with the mind I myself serve the law of God; but with the flesh the law of sin. (Rom. 7:14-25)