How does the title fit in the context of the story?

The Departed stands as a honorable and respectful way to refer to those who have passed away. As correctly explained in the first line by coleopterist.

In the movie, every action and consequence happens due to the death of somoneone, or because someone has departed.

The first scene and character, Collin Sullivan is introduced to Crime and Frank because he is now brought up by Frank (his parents have departed).

Billy Costigan is made an undercover cop by Captain Queenan and Sgt. Dignam because of his family ties to crime. The crime syndicate decide to hire him because of his family (father, uncle, et. all) - All Departed.

The Psychiatrist, Madolyn, is in business because cops come to her whenever they have killed someone (The Departed).

Queenan is killed (Departed) and this causes all hell to break loose.

Later, Frank and his entire crew are killed (All departed), and Sullivan is frantic because Frank was an FBI informant. Killing Frank, Sullivan becomes the hero of the cop force.

Later, Billy catches Frank at the building where Queenan died (the place where he departed). Officer Brown shows up, and all three go down the elevator in a stand-off. Once the doors open, officer Barrigan shoots Billy and Brown. Sullivan shoots Barrigan in return. Now, Billy is The Departed.

This earns Sullivan his keep, and no one knows that he was the real mole. However, Madolyn had heard the CD recordings and she knows what must have happened when Billy died. (again, the departed have changed the course of events).

Later when Dignam arrives, he goes over and kills Sullivan. Finally killing all the pieces of the story.

Thus, at every turn The Departed (whether people who have died earlier, or have died at that moment) change the course of the story and the characters. Hence it is a very unique way to name the film for something that is very subtle to notice, and yet has such a powerful impact on every aspect of the film. Therefore, the name The Departed.

The Departed refers to death and the dead. The movie has plenty of both.

When Costigan is at his mother's grave, he sees a wreath from Costello with a note reading, "Heaven holds the Faithful Departed". The phrase "faithful departed" appears to be used by Christians to refer to deceased members of their congregation. Considering the movie's plot dealing with identity and mixed loyalty, it is a rather ironic turn of phrase.

Just to add: The Departed was a remake of the Hong Kong movie ???. In Chinese-Buddhist religion/mythology, that is the worst hell. It corresponds to the (Sanskrit/Pali) Av?ci.

In Buddhism there are many "hells/purgatories", but Av?ci is the worst hell and usually inescapable. (In other hells, you usually get to be reborn into something better. In Av?ci you just keep getting reborn there.)

The Chinese name is also an evocative phrase that may be transliterated as the "non-stop way".

Sometimes (usually?) Hollywood remakes simply translate the original title. But in this case there is no good equivalent translation (especially since "hell" is an over-used cliché in English). So I guess they decided to settle on a similarly dark title like "The Departed".

(P.S. The HK movie used the somewhat unremarkable English title Infernal Affairs, which has pretty much nothing to do with the Chinese title.)