A motion for leave is a motion filed in court which asks the court to consider allowing the filer of the motion to depart from the established procedures of the court, most classically procedures setting out specific timeframes which must be followed when filing documents and making motions. For example, a motion for leave might ask the court to accept a document after the filing deadline.
The decision to grant a motion for leave is at the discretion of the court. The subject of the request is not a right under the law and the court can determine that it should not be granted, given the available information. The motion usually presents information which is designed to be compelling for the court. The judge weighs the information and makes a determination, which may be offered with an opinion explaining why the motion was denied or approved.
Judges are allowed considerable discretion on the bench. While they cannot make rulings which violate the law and they must follow the rules of the court, the rules do provide room for judges to make decisions about how matters should proceed. This is done with the understanding that every trial is different and it is impossible to come up with rules for every occasion. If judges were bound by utterly inflexible rules, miscarriages of justice might result.
Essentially, this type of motion asks permission to do something. The court considers what is being asked, why the request is being made, and what the outcome of denial or approval will be. For instance, someone may file a motion for leave asking for permission to file an amicus brief after the deadline has expired. The judge may determine that the brief includes important information which is highly relevant to the trial, and thus that the motion should be granted. On the other hand, the judge might argue that the brief contains no substantially new material and thus there is no benefit to granting the motion.
Like other legal motions, a motion for leave is sometimes used as a stalling tactic. While the motion is being considered, court cannot proceed, and this may buy time to work on an aspect of a case. Judges are well aware of this and may frown upon lawyers who try to use too many stalling tactics in the course of a trial. In the United States, this can interfere with the Constitutionally guaranteed right to a speedy trial.