It is important to remember the form that this excellent and unforgettable poem was written in. Coleridge wrote "The Rime of the Ancient Mariner" as a literary ballad. The ballad form uses simple language, a good deal of repetition and strong patterns of rhyme and rhythm. Thus it is no surprise that this poem itself contains a number of examples of repetition.
To focus on two however, in lines 20 and 40, we see the same phrase repeated: "The bright-eyed Mariner." The brightness of the Mariner's eyes is something that is referred to again and again at various stages throughout the poem, and here this repetition focuses on the strange appearance of the Mariner whilst also reminding us about the form of the poem.
Secondly, in the second section of the poem, there is another good example of how repetition is used for effect:
Day after day, day after day,
We stuck, nor breath nor motion;
As idle as a painted ship
Upon a painted ocean.
Note here how the repetition in these lines reinforces the meaning. The repetition of "day" and "painted" suggests the helpless, trapped feeling the sailors experience when the ship is becalmed and they are unable to move it in any direction at all and are effectively helpless.