Who is telling the story of "The Rime of the ancient Mariner?"
At the beginning of the poem, the setting is a wedding feast or reception. There are essentially three speakers in the poem. There is a third person narrator; that is to say a narrator not in the story of the poem. The other two speakers are the Mariner and the wedding-guest. The poem begins with an introduction of the Mariner by the narrator, and this is followed by the wedding-guest speaking to the Mariner:
It is an ancient Mariner
And he stoppeth one of three.
"By thy long grey beard and glittering eye,
Now wherefore stopp'st thou me?
You can tell when the narration shifts from the Mariner or the wedding-guest back to the third-person narrator because the third-person narrator will necessarily use third person pronouns (he, she, they, it) to describe things. For example, still in Part 1, the third person narrator describes how the wedding guest is captivated by the mariner's stare:
He holds him with his glittering eye-
The wedding-guest stood still,
And listens like a three years' child:
The Mariner hath his will. (13-16)
The third-person narrator has limited insight. He (the narrator) basically fills in the gaps between the Mariner's story and the wedding-guest's reactions. There are, in a sense, three levels (or frames) to this poem. There is the third person narrator who frames the entire poem. Within that frame we have the Mariner telling his tale to the wedding-guest at a wedding reception. And within that frame/world of the wedding reception, we have the frame/world of the Mariner's adventure.
The direct answer to your specific question (who tells the tale of the ancient mariner) is primarily the Mariner. He tells his actual tale to the wedding-guest. But if the question is who tells the tale of the poem as a whole, the answer is the combination of narrator, Mariner, and wedding-guest. Although the wedding-guest is merely reacting to the Mariner's tale, his reactions do inform the significance of the poem in that the Mariner's tale is a lesson the Mariner must impart to others. In other words, the Mariner tells his tale to the wedding-guest. But in the larger frame of the entire poem, the narrator recounts the story of the Mariner telling his tale to the wedding-guest.