Jesus promised the disciples that the Holy Spirit would be the ultimate Helper, even when Jesus isn't there in person. Then Jesus goes into more detail on what the Holy Spirit does.
Convict the world of sin (16:8-9) -- the Holy Spirit alerts us that we have a "sin" problem. "Sin" is basically any action that ruins others' lives without a good cause (especially if such action is for selfish reasons), and sin keeps us from a relationship with God. Jesus's death and resurrection counters the side-effects of sin, but when people don't believe in Jesus, they don't take use of Jesus's counters for sin.
Convict the world of righteousness (16:8,10) -- the Holy Spirit reminds us that Jesus is righteous because he can go to the Father with no problem. Additionally, for those who believe in Jesus as Savior, Jesus gives the Holy Spirit, which has basically the same righteous record as Jesus, who took on our penalty of sin.
Convict the world of judgment (16:8,11) -- the Holy Spirit reminds us that Jesus provides the only way to be cleared of a bad record in judgment (because good works can't help make a record perfect if the record is bad already). Additionally, the ruler of the world (AKA the devil) has been judged already via the ruler's actions.
The Holy Spirit helps show people that Jesus provides a way to be cleared of our sin record through the death and resurrection. As noted in John 3:16 and 5:24, we get clearance and eternal life by believing in Jesus as Lord and Savior. Believing in Jesus is essentially accepting that we need Lord Jesus (who is Son) to clear us from our sin record and counting on Jesus and his resurrection for providing us His righteousness. That's it for us. Jesus as Lord (and Son) rules over death and shows us how to be saved and Jesus as Savior does the saving itself. When we believe in Jesus, we are guaranteed everlasting life that would ultimately not end.
For the readers who decided to accept Jesus and count on Him for saving us, you are guaranteed eternal life. I don't have a "I raise my hand" button or "I accepted Jesus" button because faith starts internally. (You can raise your hand though.)