Yeah, if the police detain you to a drunk tank, you've been arrested. If they just think you're being a nuisance for being intoxicated, they issue a 'public intox' ticket. Either way, courts and fines are in your future. Interactions with police rarely ends with out some sort of dip into your wallet.

Fun fact, if you are arrested while in possession of any kind of narcotics, most departments will place you in the drunk tank as well until you are arraigned.