Can criminal cases be appealed?
Yes, the defendant may appeal after she or he is convicted at trial. Convicted defendants often do appeal their convictions.
If the defendant is acquitted (found "not guilty") at trial a prosecutor can not appeal. A retrial would constitute "double jeopardy" which is not permitted under the Fifth Amendment to the United States Constitution. However, before trial (or during the trial in extra-ordinary circumstances), a prosecutor generally can appeal certain matters, such as a ruling that would "suppress" or refuse to allow evidence to be presented at trial.
For example, if there were video-tapes of O. J. Simpson murdering Nicole Brown Simpson that the judge refused to allow the jury to see, the prosecutor possibly could have sought to appeal the judge’s ruling denying use of the evidence right then and there. But after the acquittal the prosecutor could not have appealed and asked for a new trial on the grounds that the tapes were improperly excluded from the trial.
Similarly, if immediately after the jury's acquittal of Simpson the Los Angeles District Attorney's Office first discovered such video-tapes, the prosecutor could not have sought a retrial on the grounds of newly discovered evidence. On the other hand, if the jury found Simpson guilty in the criminal case you can bet there would have been an appeal.