An alibi was an explanation of one's whereabouts or activities during the commission of a crime.
Such an explanation was usually given to a detective with a police force who was investigating a crime. If other evidence was found which supported such an alibi, the police then investigated other individuals. If there was no other evidence, the police investigated such an individual more closely.
During the investigation of the supposed murder of Ibudan, Odo's alibi failed to clear him. He had assumed his natural state in his security office, with no witnesses. The knowledge of this was used by Ibudan in his attempt to frame Odo for his murder. (DS9: "A Man Alone")
In 2365, while investigating a murder, Quark eventually confessed to Odo that Kira Nerys had paid him for an alibi for the time of the murder. Despite her disproved alibi, he did believe that she was innocent. He did not find out until five years later that Kira actually was the murderer. (DS9: "Necessary Evil")