This poem, though short, can be interpreted in many different ways.
Essentially, the speaker is on his way through snow-covered woods which he thinks are owned by a man he knows who lives in the village. The speaker notes that the man who owns the woods won't see him stopping there "to watch his woods fill up with snow." The suggestion, in this line, is that the speaker is stopping in the woods simply to observe their beauty.
Further, the speaker notes that his horse must think it's strange that they are stopped in the middle of the woods; he "gives his harness bells a shake/To ask if there is some mistake."
The poem ends with the following lines:
The woods are lovely, dark and deep.
But I have promises to keep,
And miles to go before I sleep,
And miles to go before I sleep.
This last stanza can be interpreted in many ways. Some read it literally--meaning that the speaker, though he enjoys the beauty of the woods, he must keep going, because he has things to do and a long distance to cover until he is able to sleep.
In a darker interpretation, some feel this poem is about the speaker's contemplation of suicide. Because he's in the woods on the darkest evening of the year, and because the horse knows something is wrong, many readers interpret the last stanza to be the speaker's ultimate decision against the suicide he was contemplating. In this interpretation, "sleep" means "die," and the "miles to go" references the life experiences the speaker still has ahead of him.