Why has the Spirit come into this world? What does God's Holy Spirit want to do in the life of the unbeliever? The Holy Spirit is very involved in the actual work of conversion. You see, before we were Christians, it was the Holy Spirit who convicted us of our sin (see John 16:8). Another way to translate the word "convict" in John 16:8 is "convince."
Notice this verse doesn't say He will convict the unbeliever of a specific sin. Rather, He wants to convince him or her of sin in general, the root cause of all sins.
Now we can try to produce in someone a sense of guilt and wrongdoing. In an effort to "help" the conversion process along, we want to make them feel guilty or to feel really bad about something. Mothers seem to have an unusual ability in this area. But only the Holy Spirit can effectively produce a guilt that will bring a person to their senses.
Sometimes we get in the way of conversion. We get impatient, or we try to assist the Spirit. We can be telling someone about the Lord, maybe a friend or a coworker or a family member, and as they are getting interested and asking questions, we see they are getting closer. So we start trying to convert that person in our own strength. We try to complete the transaction when the Spirit is still working.
The best thing we can do after we have shared the Word of God with someone is to pray that it takes root. We should just do our part and leave it in the hands of God. We don't need to force the issue. He will convince a person. Let God's Spirit do His work.