He preached to the spirits in prison. (3:19), The gospel was preached to dead people. (4:6)
Apparently these verses refer to the harrowing of hell, where Jesus preached to dead people in hell.
I wrote this short letter with Silvanus. (5:12)
Silvanus is mentioned several times in the Pauline epistles (2 Cor 1:19, 1 Th 1:1, 2 Th 1:1). Some apologists claim that he was a Greek-speaking Christian, and that he helped compose and translated Peter's letter into Greek.
The church in Babylon sends you greetings, as does Mark. (5:13)
"Babylon" here is the codename for Rome, since the Jews old enemy was Babylon and now Rome is the enemy of the Christians.
Mark (Marcus in the KJV) may refer to John Mark, who is mentioned in Acts (12:12, 12:23, 15:37-39) and 2 Timothy (4:11). I suppose this could be the same guy.