Following a hung jury, the judge declares a mistrial. The prosecutor has the option of filing the charges against the defendant again.
The prosecutor also has the option to drop the charges altogether. If the case is retried though, a new jury hears the case. Retrying the case on the charges that the jury was unable to reach a decision on does not constitute double jeopardy. Double jeopardy only applies if the jury was able to reach a "not guilty" verdict.
If the defendant was facing multiple charges, and the jury reached a guilty verdict on any of those, sentencing occurs on those charges.