In The Rime of the Ancient Mariner, to whom does the mariner tell his story?

The mariner tells his story to a random younger guy who's on his way to attend a family wedding. This guest is actually outdoors and on his way into the wedding, and is a stranger to the mariner, who stops him to tell him the tale. The speaker of the poem calls this listener "The Wedding-Guest."

I was able to answer this question by looking closely at the first few stanzas of the poem. Let's take a look:

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?

That first stanza is saying that an old mariner (a sailor) sees three people and stops one of them. And the person who was stopped says, approximately, "Hey, what's going on? You've got a long beard and an intense look in your eyes. Why did you stop me?"

I could tell that the third and fourth lines of that first stanza were spoken out loud because of the open quotation mark before the word "By." You can also tell that the person is going to keep talking in the next stanza, because there hasn't been another quotation mark yet to show that the speech is over. Let's see what else this wedding guest says:

The Bridegroom's doors are opened wide, 

And I am next of kin; 

The guests are met, the feast is set: 

May'st hear the merry din.' 

Okay, so the guest keeps talking all through this second stanza, and we can tell that he's done talking at the end, because of the quotation mark after the word "din." Here, the guest is saying that the groom's doors are open, that the guest is closely related to the groom, that all the other wedding guests are already there, that the food is ready to be eaten, and that you can even hear from outside how happy and loud everything is inside the building where the wedding is being held.

Here's the third stanza:

He holds him with his skinny hand, 

'There was a ship,' quoth he. 

'Hold off! unhand me, grey-beard loon!' 

Eftsoons his hand dropt he.

So here, the old guy has grabbed onto the wedding guest's hand and launched straight into a story about a ship. But the wedding guest is freaked out and says, approximately, "Quit it! Let go of me, you crazy guy." The old guy lets go of the young guy's hand.

Let's check out one more stanza:

He holds him with his glittering eye— 

The Wedding-Guest stood still, 

And listens like a three years' child: 

The Mariner hath his will. 

Here, we see that although the mariner has let go of the wedding guest's hand, the mariner is still "holding" onto the guest by the power of his sparkling eyes. So the guest stands there and, like a little kid, listens obediently to the story.

We can be sure that it's this same "Wedding-Guest" who listens to the whole tale because he's mentioned at the end of the poem, too: the speaker of the poem tells us that the wedding guest, having listened to the whole story, is stunned and sad but also a bit wiser than he was before he heard the mariner's tale.