There are several aspects of this symbol in the novel. We will hit two big ones here. First, the most obvious: the cross is a symbol of Christianity, of the crucifixion of Christ. We first see the symbol evoked directly by Marmeladov, who speaks these famous words:
"I ought to be crucified, crucified on a cross […]! Crucify me, oh judge, crucify me […]." (1.2.36)
Essentially, this is a desire to bring meaning to his suffering. In Christian theology, Christ's suffering on the cross is meaningful because it serves the greater good – that is, it absorbs sin. Yet, it is only because Christ himself is not subject to sin that he can be a container for the sins of regular sinning people.
Marmeladov would be considered a sinner in terms of Christian theology. So, he can't take on the sins of the world, like Christ can. Yet, his suffering is proof that he is not completely lost, that he is not beyond redemption.
See, if he was enjoying the fact that his daughter was a prostitute, or the fact that his wife and children are living in abject poverty, or any other aspect of his horrific life, that would mean that he was lost. So long as he knows he's a sinner, and wants to suffer as much as possible (as much as Christ, the ultimate sufferer) then there is hope that his soul can be freed from sin in the afterlife, thus bring meaning to his suffering.
Raskolnikov, in his mind, accuses Dounia of a similar (but different) sentiment after he reads of her plans to marry Luzhin. That's what he means when he says, "Bitter is the ascent to Golgotha" (1.4.1). Golgotha is believed to be the city where Christ was crucified. Raskolnikov accuses Dounia of taking the martyr's path, of sacrificing herself for his own sins. Ironically, Raskolnikov kills the pawnbroker, in part, to free Dounia from the need to crucify herself for him.
Sonia's position adds much complexity to this discussion. She tells Raskolnikov, "We will go to suffer together, and together we will bear our cross!" (5.4.163). Whoa. Check out that exclamation point. Sonia is excited about this idea. Like her father, she considers herself a sinner. She sees the suffering of life in Siberia as a way to stop sinning (i.e., stop being a prostitute). Even though Andrey Semyonovitch has tried to convince her that prostitution is not a sin, she doesn't believe it.
In fact, Sonia is right about her decision to join with Raskolnikov. It does allow her to start a new life, where she is accepted by the community, and where she doesn't have to resort to prostitution. For Sonia (in contrast to her father) the cross is a symbol of redemption not only after life, but also during life. As such, it also functions for Sonia as a symbol of engagement, as we see in this passage.
Without a word Sonia took out of the drawer two crosses, one of cypress wood and one of copper. She made the sign of the cross over herself and over him, and put the wooden cross on his neck." (6.8.10)
Some people use rings to get engaged, Sonia use crosses.
This brings us to the other aspect of crosses we want to look at. If you search the online text of Crime and Punishment for the word "cross" you are going to learn that Raskolnikov is constantly described as crossing the street, or the bridge, or the market. Furthermore, a driving force behind everything he does is his desire to cross over into new realities.
We get lots of these kinds of crossings in the novel. That's why we used Sonia and Raskolnikov's engagement to transition to this aspect of the discussion. Birth, death, engagement, marriage, conversion to a new religion, committing murder – these are all examples of crossing over from one kind of reality to another. The tension between the Christian idea of the cross, and the idea of crossing over into new experiences, sometimes by breaking through societal boundaries helps maximize the complexity factor in this novel.