What happens in the discussion between Hamlet and the Gravedigger? How does the mood differ from the previous scene?

The overall mood of act 5, scene 1 is a good deal lighter than the previous scene. There, we were informed of Ophelia's tragic suicide by drowning. A pall of unremitting gloom hangs over the royal palace as everyone tries to deal with their grief over such terrible news.
The following scene immediately establishes a radically different tone. A couple of clownish gravediggers engage in humorous word-play and witty banter as they prepare Ophelia's final resting place. When Hamlet arrives on the scene, he enters into the general spirit of levity, albeit with a dark, mordant wit that is far removed from the absurd badinage of the gravediggers. Hamlet doesn't sing at the graveside like the clowns, but he does still display a nice line in black humor as he muses sardonically on the life of the courtier whose skull he might possibly be holding:

Or of a courtier, which could say, “Good morrow, sweet lord!” “How dost thou, good lord?” This might be my Lord Such-a-one that praised my Lord Such-a-one’s horse when he meant to beg it, might it not?

The tone is humorous but more biting than that of the gravediggers with their clownish tomfoolery. Nevertheless, Hamlet's irreverent barbs at the high-born dead, now brought low by the ultimate leveler, death, provide a welcome respite from the tears and tragedy of the previous scene.
Both brands of humor are on display when Hamlet engages in conversation with one of the gravediggers as he tries to establish the identity of the person about to be buried. But obtaining information from the gravedigger is difficult, and this in itself adds to the general air of humor.
The mood changes slightly, however, when the gravedigger identifies the skull of Yorick, the former court jester. This causes Hamlet to embark upon one of his most famous monologues, in which he muses on the common fate of all, from the highest born such as Alexander the Great, to the lowliest clown like poor Yorick.