Conviction comes before conversion. It is conviction by the Holy Spirit. Left to ourselves, we would never turn to God (Romans 3:10-12). The Holy Spirit convicts us of one particular sin above all others, not believing on the Lord Jesus Christ (John 16:8-11). Conviction in itself is not conversion, but it is the Holy Spirit's work before you are converted.
Conversion is repentance toward God and faith toward our Lord Jesus Christ (Acts 20:21). Repentance and faith go together like the two sides of the same door. Repentance is turning away from our sins, in particular the sin of unbelief. Faith is turning to God in Christ Jesus as our Lord. Repentance and faith are one and the same action in conversion.
Without conviction by the Holy Spirit, no one is converted. The Spirit of God convicts us, before we repent and believe on the Lord Jesus Christ. That's the reason Millard Erickson could say, "Without the work of the Holy Spirit, there can be no conversion."