She walks in beauty, like the night Of cloudless climes and starry skies; (lines 1-2)
The opening simile of the poem compares the unnamed woman to vast and intangible things, like "night" and "starry skies." Why can't he just compare her to a flower, or to something that we can wrap our minds around more easily?
And all that's best of dark and bright (line 3)
If you thought "night" and "starry skies" were abstract ideas, that's nothing. In this line, the speaker relates her beauty to "the best of dark and bright." How can you have a "best" (or worst) of darkness or brightness? We're not used to making value judgments about things as abstract as light and dark.
Thus mellow'd to that tender light Which heaven to gaudy day denies. (lines 5-6)
More vast and intangible things! The woman now seems to almost be casting a gentle glow ("tender light"). Her glow is much better than daylight, though – "day" is described as "gaudy," or over-bright and garish, by comparison.
[…] the nameless grace (line 8)
The woman remains "nameless" in this poem, so it makes sense that the speaker can't find the words to describe her "grace."
A mind at peace with all below, (line 17)
The woman is "at peace with all below." With everyone on earth? Really? She has no outstanding grudges or disagreements with anyone? This makes her seem almost saintly – no wonder the speaker is in awe of her.