So, basically, what does happen when someone gets arrested?

In Innsmouth Horror there's a whole system for it, but in Arkham Horror? Does the investigator simply moves to the police station and ends his turn?

Galaahd said:

You get moved into the jail cell portion of the police station and lose half your money rounded down. You then lose your next turn, standing your investigator in the actual police station during the movement phase, but nothing more. The turn after that, you are free to go.

You go to the jail, lose half your money and end of turn.

Next turn, you go to police station and finish your turn ? You don't have an encounter at police station ?

My group plays that you get an encounter at the police station after you get out of the cell (similar to being delayed)

I'm not sure if that's correct technically but it does give Deputy Dingby another chance to mess up your investigation

Correct, you do not technically get an encounter at the police station, since when arrested you lose your next turn (meaning you also skip the upkeep phase, in addition to the encounter phase),

However, giving someone an encounter can be sorta neat, and also prevent that character from being bored. Then again, that's also part of the punishment, not getting a turn, or the chance to possibly get items or clues...