In "The Love Song of J. Alfred Prufrock" who/what do the mermaids represent, and why does Prufrock not think that they will sing to him?

The answers to your question lie, in part, directly in the
context of the lines you refer to in Eliot's "The Love Song of J. Alfred Prufrock." 
Here are the lines:

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Shall I part my hair behind [to cover a bald
spot?]?  Do I dare to eat a peach [because he wears
dentures?]?

I shall wear white flannel trousers, and walk
upon the beach.

I have heard the mermaids singing, each to
each.

I do not think that they will sing to me. 
(lines 122-125)

First, the
mermaids will not sing to him because he's old.  He is not socially gifted, and his
prime, if he ever had one, is long past.

The mermaids can
probably be identified by other lines that follow those
above:

I have
seen them riding seaward on the waves

Combing the white
hair of the waves blown back

When the wind blows the water
white and black.

We have lingered in the
chambers of the sea

By sea-girls wreathed with seaweed red
and brown

Till human voices wake us, and we drown. 
(126-131)

The
mermaids are an allusion to the sirens of Ancient Greek myth, who
would sing to lure sailors to their underwater caves, then stop singing (breaking their
spell), thereby drowning the sailors.  The sense here is that the speaker daydreams, and
then is awakened from his daydreams by someone talking, since the voices are "human
voices."  This, of course, connects the mermaids/sirens to the women with, or, I should
say, in the room, with Prufrock.

But neither the women, nor
the sirens, who are not particularly picky about the men they drown, will sing to
Prufrock.