Your sentence is the neutral way to say it. "Egy diákot keres a bíró." puts the emphasis on "diákot".
What, like in English phrasing it: "It's a student that the judge is looking for." ?
This is the accusative case, which shows who or what is it that's directly affected by the action.
In English you mainly use the word order to express this: The dog bites Peter isn't the same as Peter bites the dog.
Hungarian on the other hand uses a -t suffix to express this, so the word order doesn't matter. So using the same example: A kutya megharapja Pétert and Pétert megharapja a kutya both express that it's Peter who gets bitten.
And it just happens that keres is a verb that requires the accusative case, unlike its English counterpart which requires the preposition for. Differences like this simply have to be memorised.
Thanks for this explanation. It's like some lights have just been switched on for me.