This little ditty only has three stanzas, and at the top of each one the speaker compares—you guessed it—the brain to, well, something else. So, we figure that this lump of grey matter must be of above average importance to the poem. All in all, it represents more than just the literal organ in our skulls, it represents all of human knowledge, as well as our ability to learn and synthesize information.
The poem kicks off by announcing proudly that "The Brain—is wider than the Sky—." Assuming that the speaker isn't a lunatic with a very poor understanding of anatomy, this is meant metaphorically. It's Brain with a capital B, emphasizing that it represents all those little brains out there.
More than that, it doesn't just represent the lumps of flesh we call brains, it represents the human intellect that those lumps of flesh mysteriously house. When you think about all that human consciousness out there—like really think about everyone who's alive and all the things they know—it really is immeasurable, which might just make it wider than the sky.
In the second stanza, the speaker continues her rant on the awesomeness of brains by saying they're "deeper than the Sea." All the same stuff applies here that we said above about the immeasurable human consciousness and all that. Like the sky, the sea seems super deep when you look into it, but when you try to truly look into the depths of your or someone else's mind, it seems impossibly deep as well.
We just want to point out the neat trick Dickinson pulls here by comparing the brain absorbing knowledge to a sponge absorbing water. Brains look a whole lot like sponges, right? That's it. That's all we've got on this one.
The speaker saves the biggest boast about the brain till the end, when she says that it's "just the weight of God." We think it's interesting that she chooses to compare the two things in terms of weight specifically.
Could it be a metaphor for way some people feel weighed down by guilt over what they see as their sinful lives? Could it be talking about the burden of consciousness, that sometimes our over-thinking brains make it harder to be alive? What does your brain tell you?