The speaker moves back and forth between her own grief and that of others. Recognizing the pain of others is kind of a big deal for her, because it gives her comfort and a sense of (imagined) community. Ultimately, though, it's her own pain that is the real motivation for the poem.
Lines 3 and 7: The word "mine" is repeated at the ends of these lines. Excuse us while we get all fancy here, but the repetition of phrases at the end of a line is known as epistrophe (as opposed to anaphora, which occurs at the beginning). Dickinson's use of epistrophe here (combined with the fact that "Mine" is capitalized both times) highlights the speaker's pre-occupation in her own suffering.
Stanzas 1 and 2: These stanzas have the same pattern—the speaker begins by speaking about others, then compares them to herself. It's almost as if she's using what she infers about other people's pain in order to get to know her own pain more.
Stanzas 7 and 8: Here's a handy list of different kinds of grief—just in case you ever get confused.
Lines 1 and 40: If you have any doubt about the overall focus of the poem, remember that the first word of the poem is "I," and the last words are, "My Own." It's subtle, but paying attention to the number and kind of pronouns in a poem can lead to important insights, and be handy reminders of its real subject matter.