What did Robert Burns promise in the poem "A Red, Red Rose"?

In "A Red, Red Rose," the speaker uses two types of figurative language to express his love. First he uses hyperbole, a type of figurative language that exaggerates for emphasis or humor, to declare his love to his "bonnie lass," or beautiful girl. Second, he uses a metaphor, which compares one thing to another without using the words "like" or "as." Four separate promises are made in the poem.

1. "And I will luve thee still, my dear, / Till a’ the seas gang dry." To translate this into everyday speech, it means, "I will love you until all the seas dry up." Since the seas will never dry up, this is hyperbole, an exaggerated way of saying, "I will love you forever." 

2. I will love you until "the rocks melt wi’ the sun" means "I will love you until the heat of the sun melts rocks on earth." Again, this will never happen, so it is another hyperbolic way of saying the speaker will love his girlfriend forever.

3. "While the sands o’ life shall run" is a metaphor comparing one's lifetime to an hourglass. If all the days of his life are like sands in an hourglass, he will love his girl until the very last one. He will love her until the day he dies.

4. "And I will come again, my luve, / Though it were ten thousand mile" is another hyperbole. This means that, although he must leave his love now, he will return to her, even if he has to travel ten thousand miles to do so. Nowadays someone might actually be separated by ten thousand miles from a loved one. For example, the distance from Scotland to parts of Australia is 10,000 miles. When Robert Burns was writing these poems in Scotland in the 1700s, not many people would travel that far from home, but the literal distance isn't really the point. Using such a large round number means the same as saying, "no matter how far I have to go." So the speaker is promising to return despite the distance that will be separating them. 

Using hyperbole and metaphor, the speaker promises to always love his "bonnie lass" and to return to her no matter how far away he must go from her side.