The poem begins with the speaker making a series of urgent requests. The speaker wants to stop “the clocks,” to turn off the “telephone”, to give the dog a “juicy bone” to keep it from barking, and to “silence the pianos.” Many of these requests are symbolic. For instance, when the speaker asks to “stop all the clocks,” the speaker is really asking to stop time itself. The telephone, meanwhile, might represent modern life and business: the speaker isn’t asking to “cut off” a specific telephone, but all telephones, and with them their constant stream of interruptions and information. Finally, the “pianos” symbolize raucous parties and celebration. Overall, then, the speaker is asking for a moment of peace and stillness, free from distracting noises, a moment of somber reflection.
In lines 3-4, it becomes clear why the speaker wants this: someone important to the speaker has died. (It's possible to read this as a literal death, or as a metaphor for the end of an important relationship.) The speaker is asking the rest of the world to mourn with the speaker, to acknowledge the magnitude of this loss. But the fact that the speaker has to ask for the clocks to stop and the telephones to be cut suggest that the world hasn’t stopped to accommodate the speaker’s grief. In other words, there’s a disconnect between the world and the speaker. The speaker is heartbroken, and the world seems indifferent to the speaker’s grief: it keeps moving along, business as usual. As such, the speaker feels isolated and demands that the world slow down, stop, bring itself in line with the speaker's grief.
The first four lines also establish the poem’s form. The poem is written in quatrains: it has four stanzas, each with four lines. Each quatrain is composed of two rhyming couplets (creating an AABB rhyme scheme). The poem uses meter, but its meter shifts around unpredictably: lines 1, 3, and 4 are in iambic pentameter, while line 2 is probably best thought of as being in iambic hexameter. Recall that an iamb is a poetic unit with a da DUM stress pattern; pentameter has five iambs per line, while hexameter has six.
There are also lots of metrical substitutions throughout. The first line, for instance, opens with a spondee (stressed-stressed) or a trochee (stressed-unstressed), depending on how it's read; either way, this adds extra emphasis to the speaker's request:
Stop all | the clocks, | cut off | the tel- | ephone
Prevent | the dog | from bark- | ing with | a jui- | cy bone,
Silence the | pia- | nos and | with muf- | fled drum
Bring out | the cof- | fin, let | the mourn- | ers come.
As is clear above, the meter definitely isn't regular. Line 3, for instance, actually opens with a dactyl (stressed-unstressed-unstressed). Altogether, these shifts in the meter add a sense of instability to the poem's rhythm.
The poem almost exclusively uses end-stopped lines. This creates a sense that each line is cut off from the next, a sense which is strengthened by the caesuras that divide lines 1 and 4. Overall, the poem’s sentences and phrases tend to be discrete. The speaker doesn’t use coordinating words like “and” or “therefore” to show the reader how each is related to the next. Instead, the reader has to figure out for themselves how everything fits together. This is an example of the poetic device asyndeton—a device the speaker uses throughout the poem. Without such clues about how the speaker’s ideas and requests are related to each other, the poem feels spontaneous and unplanned—an overflowing of grief that expresses itself in a turbulent, sometimes disorganized flow of thoughts.
The speaker does give some hints that the reader should try to assemble these discrete ideas. For example, the speaker uses an alliterative /k/ sound in lines 1 and 4, in words like “clock,” “cut,” “coffin,” and “come.” Bracketing the stanza, these alliterations suggest an underlying continuity that runs through the speaker’s grief. And, at times, the speaker breaks the poem’s pattern, slipping in an enjambment—as line 3. This enjambment falls at a crucial point in the poem: the first time the speaker admits that someone has (metaphorically or literally) died. The idea is so upsetting for the speaker that it causes the poem to skid a little, to lose its confidence.