Different Ways for Arrest to Occur
Warrantless Arrest - Under certain situations cops can make arrests without a warrant. They are allowed to make arrests if an alleged offense is committed in the officer's presence or within the officers' immediate knowledge, or if the alleged offender is trying to escape, or if the officer believes that an act of family violence occurred, or "for other cause there is likely to be failure of justice of want of a judicial officer to issue a warrant.
So, the sad reality is that cops can essentially arrest anyone, at any time, for anything without a warrant if they are willing to sign their name to an affidavit saying it happened.
Warrant for Arrest - A lot of times officers do get a warrant for someone's arrest. This happens when an officer goes to a judge and explains to them their side of the story as to what happened (of course we don't get to go and present our side). Once that arrest warrant is issued the warrant will be placed into a system maintained by the GBI and eventually you will be arrested on that warrant.
Grand Jury Indicts - Sometimes, often in serious cases, there will be no warrant issued until a grand jury considers a case. More on grand juries here. When a grand jury finds probable cause, and they almost always do because, once again, we can't go and present our side of the story. A judge will sign an arrest order that orders you be arrested.