Drink to me only with thine eyes And I will pledge with mine;
The poem opens with the speaker addressing a woman (presumably). Because of the title, we're guessing she's named Celia.
He tells her to "drink" to him "only" with her "eyes." In other words, he's telling her that she doesn't have to hold up a beer and say cheers, but only has to use her eyes. It's kind of like when you say, "I'll drink to the Bears winning the Super Bowl."
The speaker says that he, too, will "pledge" – i.e., "drink" or say "cheers" or something to that effect – with his eyes.
Or leave a kiss but in the cup, And I'll not look for wine;
If she doesn't want to "drink," the speaker says she can just leave a kiss "but in the cup," and he won't care if there's no wine in it ("I'll not look for wine").
"But" in line 3 means something like "just" or "only," and it sounds funny because it's out of place. The line really means "just leave a kiss in the cup, baby" or "only leave a kiss in the cup."
The thirst that from the soul doth rise, Doth ask a drink divine:
The speaker explains his desire for a drink that is not a drink. You know, he's asking for a cup with a kiss in it instead of wine.
He says that his thirst isn't a bodily thirst (in other words, he's not dehydrated in the desert and craving water), but rather a more spiritual one (it is a thirst "from the soul").
Because his "thirst" is from the "soul," it requires something more "divine" than, say, "wine" to satisfy it.
Notice that line 6 is a shorter line than the previous five; they all contained eight syllables, while line 6 only contains, fittingly, six syllables.
But might I of Jove's nectar sup, I would not change for thine.
The speaker next delivers a couplet (a pair of lines) that is meant to express how he feels about the refreshing spiritual beverage Celia can offer him.
He says that even if he could drink nectar from Jove's cup ("might I of Jove's nectar sup") he wouldn't; he would rather have Celia's cup ("thine").
While this is the gist of these lines, the word choices are a bit awkward. The speaker seems to be saying "if I were allowed to drink Jove's nectar, I wouldn't 'change' the way things are, 'for' I prefer your beverage, my lady."
It is also possible that the speaker means he wouldn't exchange Jove's nectar for Celia's "nectar" of love, an interpretation that contradicts a lot of what the speaker has been saying.
Hold up! Who the heck is Jove? Good question. Jove is another name for the Greek god Zeus (or Jupiter to the Romans), the king of gods who live on Mount Olympus. The gods on Olympus are big fans of drinking "nectar."