The poem begins with a metaphor. The speaker describes "measur[ing]" everyone else's sorrow against her own, implying that she's trying to figure out whose grief is greater. She wants to know whether others are burdened as heavily as she is, or whether their sorrow is of "an Easier size" (i.e., not as difficult to live with). The fact that she does this "with narrow, probing Eyes" suggests that it isn't something she's forthright about; rather, this is a personal, internal obsession for her.
Right away, the poem creates musicality and rhythm through sonic devices: there's consonance (the /z/ sounds in "weighs," "Easier," and "size," for example), assonance (the short /eh/ sounds in "measure" and "every," the long /ee/ sounds in "Grief" and "meet"), and alliteration (the /m/ sounds at the beginnings of "measure" and "meet," as well as the /w/ sounds in "wonder" and "weighs").
The variety of sounds happening within and across lines gives the poem a rich, subtle texture that reflects the alertness and subtlety of the speaker's mind. This is a speaker who's constantly "measur[ing]," "probing," and "wonder[ing]." She isn't just examining other people's grief; she's attentively investigating the nature of grief itself.
The poem also uses common meter: its lines alternate between eight and six syllables apiece in iambic (da-DUM, da-DUM) rhythm, and follow an ABCB rhyme scheme. Here's an example of that meter in action in lines 1-2:
I mea- | sure ev- | ery Grief | I meet
With nar- | row, pro- | bing, Eyes —
Note that line 1 is iambic tetramter (four iambs, for a total of eight beats) whereas line 2 is iambic trimeter (three iambs, for a total of six beats). This meter gives the poem a feeling of formality, in keeping with both its grief theme and its speaker's reserved, observing tone. This first stanza also contains a clear, full rhyme between "Eyes" in line 2 and "size" in line four.
Finally, Dickinson's signature em dashes, seen here in lines 2 and 3, give the poem a hesitant, fragmentary feeling. Her eccentric capitalization (another signature effect) introduces further unpredictability, while sometimes lending emphasis to certain words. For example, even Dickinson wouldn't normally capitalize a short pronoun like "it," but she does here (in line 3), suggesting that grief is so weighty that it's not an "it" but an "It."