In line 3, when the narrator describes the eagle as "ring'd with the azure world," he means that the eagle is surrounded with or encircled by the bluest of blue skies: azure is the name of this pristine sky blue color, the blue we see on a perfectly sunny and clear day.  The eagle is perched very high up in the air, "clasp[ing] a crag" that is "close to the sun" (lines 1, 2).  He is alone and holding on to a tall mountaintop.  Thus, it seems as though we are sort of seeing the world the way the eagle sees it from high up on his mountain perch: the eagle perceives himself as being at the center of this blue circle of sky around him, and so this is what we see from the description as well. 

What does Tennyson mean by "azure world" in "The Eagle"?

Lord Alfred Tennyson's poem "The Eagle" speaks to a dying eagle which falls into the sea beneath him. The poem is a short six line poem, broken into two stanzas. Each stanza is called a tercet (meaning three lines). The rhyme scheme is a simple rhyme scheme (aaa / bbb).

As for the line in question,

Ring'd with the azure world, he stands,

the line speaks to the halo of the blue sky which crowns the eagle. Azure means a bright blue color and ring'd refers to the way the sky encircles the eagle's head. Essentially, the eagle's head is crowned with the sky (showing his power and importance).

The pride of the eagle is inferred in the first few lines through the clasping of the crag with his crooked hands (personification). The eagle has come close to the sun in lands which are empty (alluding to those who leave when close to death). The importance of this lies in the "crowning" of the eagle, by the sky itself (the place where the eagle rules), and the royal color which encircles his head.