The word exhibit is normally a verb, but in the legal profession (as in the art world), it is noun referring to any objects or documents (written or otherwise) that are introduced to the judge and jury as evidence of wrongdoing.
An exhibit may also be a copy of a paper that is attached to any legal document that may be filed in a civil action, which may be referred to and incorporated into said document.
Exhibits (identified by letter, as in "Exhibit A," etc.) may consist of nearly any kind of physical, documentary or digital evidence that is pertinent to the case at hand. Common examples are an alleged murder weapon (such as a gun, knife or bludgeon), an invoice or receipt for a purchase, sound recordings, a photograph or digital image, or email records.
An exhibit may be subject to objection by opposing counsel. The judge in this case has a great deal of discretion, but may disallow an exhibit if in his/her opinion it was obtained illegally, was not brought up during the discovery process, or is not relevant to the case at hand.
For example, a photograph of a man and woman having an affair taken by a police officer without any legal authorization would be inadmissible under those grounds. On the other hand, if the same photo was taken by a private investigator hired to do so by a suspicious and aggrieved spouse, the photo may be admitted into evidence, since the private investigator is not bound by the same regulations that cover police work.
Another example is the recent wiretapping by the federal government. The president may tap the phone conversation of a citizen for no more than three days before he must go to the FISA court for authorization; however, the FISA court has never been consulted. The danger now is that even if these wiretaps do contain evidence of treason or intension to carry out a domestic attack, they may be inadmissible during trial because they were illegally obtained under current law.