In what two ways is this quotation significant: "seems, madam! nay, it is, I know not 'seems'" (1.ii.76)?
Gertrude has pointed out to Hamlet that death is a natural part of life, and Hamlet agrees. She asks, and I paraphrase: So, if you know death is common, why does it seem to be affecting you so particularly? He is irritated, firstly, that she says "seems" because there can be a big difference between what something looks like and what it is. He is genuinely this upset about his father's death. He is particularly affected; he doesn't seem to be.
Then, he goes on to list the trappings of mourning -- dark clothing, tears, sighs -- and he says that a person could fake grief by using these things if he wanted to. We know that Hamlet is angry at his mother and stepfather/uncle for marrying and especially for marrying so soon after his father's death. In saying this, he is implying that her grief really was not sincere; she may have worn black and cried a lot, but the haste with which she remarried proves that she didn't truly grieve Hamlet's father that sincerely. With her, it "seemed" as though she grieved heartily for a while; with Hamlet, he truly does grieve still.
The first meaning would be literal: Hamlet is objecting to his mother saying that he appears to be upset, and is insisting that he is.
The second meaning depends on the line following: it is a pun. He says it isn't his "inky cloak" that makes him look that way—it isn't just the "seams" of his mourning clothes that make him seem upset (not just how he is dressed). It is how he is: the man, not just the clothes.
this gets to a central issue (and question) in the play:
illusion vs. reality
later on does hamlet SEEM to be crazy, as he tells horatio he will be putting on an "antic disposition" shortly after this passage (which is illusion). or IS hamlet actually crazy? (reality)