Who does Macbeth send to carry out the murders and what happens during the attempt in Shakespeare's Macbeth?

[I am assuming you mean the attempted murder of both
Banquo and Fleance in Act III and not the murder of both Duncan and his grooms.  I
suppose the latter would give you quite a different answer: 
Macbeth.]

Quite simply, Macbeth sends two "murderers" to
carry out the deed of murdering Banquo and Fleance.  (The two are literally called
"First Murderer" and "Second Murderer.")  Here is Macbeth asking them to do the
killing:

The
moment on 't; for 't must be done tonight, / And something from the palace; always
thought / That I require a
clearness.

Macbeth gives them
clarity by asking them to know that "Banquo was your enemy" and mentioning "Fleance his
son."

Of course, the devil is in the details here, for as
you suggest in your question, something goes awry when the murders are carried out.  All
we are told is that "near the palace" the murderers set upon Banquo
and Fleance.  "They set upon Banquo," says the stage directions. 
With Banquo near death he yells, "Fly, good Fleance, fly, fly, fly!"  Then comes the
rub:  Exit Fleance.

Fleance, then,
gets away, ... therefore allowing the Witches' prophesy to ultimately come
true.