This is the pair where the wrong word is most likely to offend, and the mechanism is worth understanding. A disinterested judge is doing the job properly. An uninterested judge is asleep. Call the wrong one by the wrong name and you have not made a spelling slip — you have made an accusation.
An adjective meaning impartial, having no personal stake in the outcome.
An adjective meaning not interested, bored, indifferent.
Quick way to remember
Ask whether you mean "impartial" or "bored." Impartial takes "disinterested" — think of "interest" as in a financial stake, which a disinterested person does not hold. Bored takes "uninterested." If you are describing someone's fairness, never use "uninterested."