A motion to suppress evidence is a request by a defendant that the judge exclude certain evidence from trial. The defense often makes this motion well in advance of trial—if the defendant wins it, the prosecution or judge may have to dismiss the case. Whether dismissal is appropriate depends on how important the evidence is to the prosecution's case.
Motions to suppress evidence are most common in Fourth Amendment, search-and-seizure cases. Defendants can file them in other contexts, too—for example, where eyewitness identification is involved. A defendant might argue that the identification procedure was so unfair that the judge should bar the prosecution from mentioning its results at trial.
Example: Wallace and Poot are walking down the street. They aren't behaving suspiciously, and the police don't have any reason to suspect that they've recently committed a crime. Nevertheless, Officers Carver and Haulk pull up alongside them in a car. They order Wallace and Poot to stop, then jump out of the car and frisk both. Officer Carver finds an "eight-ball" of cocaine in Wallace's jacket pocket. The prosecution charges Wallace with drug possession. Wallace's lawyer files a motion to suppress the cocaine evidence. The judge grants the motion due to the fact that the officers didn't have reasonable suspicion for the detention of Wallace and Poot, nor any basis to search them. Since the prosecution can't prove the crime without the evidence, it moves to dismiss charges; the judge grants the motion. (See What's the difference between an arrest and a detention or "stop and frisk"?)
For information on the potential fallout of losing a motion to suppress, see If I lose my motion to suppress and plead guilty, can I appeal?