An offer of proof is a kind of motion that a lawyer may present to a judge or to the official presiding over a hearing.
It is an explanation made by an attorney to a judge during trial to show why a question which has been objected to as immaterial or irrelevant will lead to evidence of value to proving the case of the lawyer's client.
The offer provides the opposition a preview of the questions (and helps prevent surprise), but is essential to overcome the objections.
In the context of a trial or a hearing, a presiding judge may issue a ruling denying a party the right to proffer evidence.
A party may request a motion in limine (Latin: "at the threshold") made before the start of a trial requesting that the judge rule that certain evidence may, or may not, be introduced to the jury in a trial.