A dispositional belief is a belief held, but not one in a state of active contemplation. This dichotomy of beliefs is generally known as dispositional and occurrent belief in philosophy of mind.
From a philosophical perspective, "disposition" is related to the philosophy of mind and questions of agency. Insofar as the philosophy of law is concerned, it is related to being culpable. For instance, in the Anglo-American tradition of law, mens rea is a frequently necessary requirement for a finding of guilt. A disposition is nothing more than a pattern of observable behavior. If you're a native English speaker, you'll likely be more comfortable with the word predisposed which is a strong synonym. Let's take an example:
An inveterate criminal is predisposed to breaking the law; whatever the origins of such a disposition, society simply cannot tolerate flouting the rule of law.
But this is not the same meaning of the same term in philosophy.
Gilbert Ryle who famously rejected the dichotomy of mind and body, had an entire chapter devoted to disposition in his Concept of Mind, a must-read if you have an interest in the philosophy of mind. Simply put, a judge has to refer to the intention of a human being when adjudicating, taking into account what the agent in question was thinking, believing, attempting, etc. but without access to that person's experience first-hand. In philosophy, awareness of your own thoughts is often described with the term 'phenomoenological', but the question arises, are we conscious of all our own beliefs? The answer is no. Enter philosophers of mind, psychologists, and forensic psychological evaluators who split fine hairs over ideas like awareness, intention, habit, consciousness, etc.
Gilbert Ryle in The Concept of Mind on p.116 (0-226-73296-7) distinguishes between dispositional and episodic behavior. The former is the tendency over time, and the latter is the immediate action. On the next page he goes on to offer that a smoker (dispositional behavior) needn't be smoking (episodic behavior). [Interestingly, in standard English, our verb doesn't draw the distinction. "She is eating" might mean he stopped fasting for his protest, or it might mean she is chewing on an apple. In the African-American dialect of English, 'he be eatin' and 'he eatin' does distinguish the dispositional from the episodic.]
Judges are interested in character and habit, so whether an action is merely episodic (he's a pacificist who shot someone in self-defense) or dispositional (he has a history of violence and has killed before) colors the case. The law is obliged to protect society from the violently dispositioned. But the first time an episode happens, it is also incumbent upon a judge to see if there is a possibility of disposition. On p.118 Ryle states "Dispositional words like ... 'believe'... signify abilities, tendencies, or pronenesses to do, not things of one unique kind...".
Later, he addresses a special subclass. On p.135, Ryle writes "[a]n important sub-class of such occurrences are those which exhibit qualities of character and intellect." He defines that occurrences, which are necessary conditions, dispositions, can be classified into 'minding' and 'absence of mind', and relies among others on the example of driving. Once can drive very carefully or absolutely recklessly or some degree between, as Conifold alluded to in his comment.
Let's offer a case:
A machine operator is given the okay by his co-worker to crush a car at the junk yard, and after pressing the button, hears a faint scream. This machine operator had been interviewed secretly by the FBI and had been advised his coworker was involved in organized crime on-site, and believed the FBI given the character and habits of his coworker he had witnessed. Is the machine operator to be charged in the crime, and if not, based on what rationale? He clearly knew his coworker was engaged in such activities, didn't bother to double check, but did directly contribute to a death. If he's not guilty of 1st degree murder, how about 2nd degree? Maybe negligence?
But what of believing? Can we differentiate between occurrence and disposition? The answer is yes. The gist of it is that a dispositional belief is one that is present in the mind, but has not been actively considered. For instance, from our car-crushing example, it is the dispositional belief that the coworker is involved in organized crime, but the belief that the coworker was presently using the equipment hadn't occurred until the scream! In this case, the first belief is a disposition, and the second belief is an occurrence. Since it hadn't occurred to the operator that he was killing a person, is he exculpated of murder, or perhaps there are elements of criminal negligence at play? The devil, as you know in case-based reason (a form of defeasible reason), is in the details. A judge would have to decide what the moral, ethical, and legal implications are of pushing that killed a person given some imperfect knowledge of the present circumstances possessed by machine operator.