"Thanatopsis" is a poetic meditation on human beings' relationship with death. The title comes from the Greek roots thanatos (death) and opsis (sight). In other words, the poem is literally about looking at death. This wasn't Bryant's original title for the poem (he didn't have one, in fact), but rather was added by the editors who first published the poem. These editors were so impressed by the poem they couldn't disbelieve that an American could have written it! This impressive sounding title captures some of that feeling. It lets the reader know that this is going to be a lofty-sounding poem, one whose rhetoric is reminiscent of earlier European masterpieces.
The poem begins by discussing the relationship between an individual "him" (i.e., anyone) and "Nature." The speaker references the ability for people to have a sacred relationship with nature:
[...] him who in the love of Nature holds
Communion with her visible forms
"Communion" is a term with Christian overtones. Most broadly, it suggests a relationship between God and the faithful. The most concrete example is when Christians consume the "body and blood" of Christ in the form of bread and wine. Here, however, there are none of the trappings of church. Instead, the speaker suggests that one can achieve a similar kind of grace simply looking by at nature. This is a form of "love."
Throughout the history of Christianity, some theologians have claimed that God is immanent in nature, making nature itself holy. That is, God is directly present in the physical world, even part of that world, rather than removed in Heaven. The speaker seems to be suggesting something similar here. This places Bryant at the start of a line of American writers, such as Ralph Waldo Emerson, who depicted the relationship between people and nature as divine (a movement called American Romanticism, or Transcendentalism).
Romantic writers, both English and American, have also emphasized how nature seems to both reflect and modify peoples feelings. That's exactly what the speaker depicts here. Nature "speaks / A various language." This personification of nature as a speaking "she" captures the sense of communion mentioned earlier—it's as if nature responds to her observers, like one friend to another. The speaking is, of course, metaphorical. For people looking at nature, the sights it offers are like responses to their feelings. And it's always just the right response. During someone's "gayer hours," that is, moments of happiness, nature seems to join in that happiness like a good friend: "She has a voice of gladness."
Furthermore, nature doesn't just mirror happiness, it also adds "a smile / And eloquence of beauty." One reason for "the love of Nature" is that nature always enriches people's experiences. Throughout the poem, the speaker will use natural imagery to enrich the reader's understanding of death.
Notice that all of this is written in the third-person ("him"). The speaker never comes out and speaks in the first-person "I." This has two, almost paradoxical effects. On one hand, instruction manuals employ a similar strategy, so that the poem can be thought of instructions for thinking about death. On the other hand, the "I" is implied. There's no "I" because it's as if the reader is inside the speaker's head, listening to the speaker's thoughts. People don't need to say "I" in their own heads! Either way, the poem has the effect of enveloping both speaker and reader in an engrossing line of thought.
The beginning of the poem also establishes the poem's meter, which will remain consistent through. This meter is iambic pentameter, or five da-DUM feet per line:
To him | who in | the love | of Na- | ture holds
The first line begins with this crystal-clear meter, which the speaker will vary dexterously throughout the poem. These line are also unrhymed, making them blank verse.