Yes. After you've been arrested, you remain in holding until a Magistrate has a hearing to officially charge you with whatever you were arrested for, set a trial date, and set bail. The general interpretation of the Constitution demands that this happens fairly quickly, as in under a week. If the police hold you longer without officially charging you and then charge you anyway it can be used as an argument in trial and upon appeal that your case was handled unconstitutionally and is therefore invalid. The longer they wait, the more likely this is to work. If the police hold you for an unconstitutional period they are probably not going to end up charging you and just want to make your life difficult.
You also have a right to a speedy trial, though this is usually waved by your defender at the Magistrate's hearing because it does more harm than good for you, as the defense needs time to build their case. Not waiving it is an option that you have a right to take, but it means that the whole thing will probably go down very quickly with neither the defense nor prosecution prepared at all to make their case. That can be a rather high-stakes gamble.
The Magistrate will then decide whether or not to set bail and what the cost of that bail will be. The Constitution demands that reasonable bail be set, though it has been interpreted by the judiciary that no bail may be set in cases where there is a reasonable belief that allowing the accused to walk free has a high chance of them hurting someone, hurting themselves, or fleeing. Usually bail is set, however, as not setting it when it should have been is another thing that can be used against the state in trial and upon appeal. The severity of the accusation usually determines the bail, small things like disturbing the peace will probably be in the hundreds, felony murder can be in the millions. Ultimately it is all up to the Magistrate, but once again, doing something like setting a 10 million dollar bail for graffiti will probably get the case thrown out of court.
Upon the setting of bail you are returned to police custody until such a time that the trial date comes or your bail is paid by your funds, or more likely a third-party. A bail bonds agent is a business entity that will post your bail for you, in exchange for a much lesser fee from you and a legal promise to attend trial. Accepting a bail bondsman's offer and then not turning up for trial is unwise.
If you are freed on bail, the state holds your bail until the date of trial and you are not allowed to leave the USA until the trial ends. If you show up, the money is returned to you. If you don't, the state takes the money and a warrant is issued for your arrest. Depending upon state law, also screwing over a bail bondsman with this allows them to send a private bounty hunter to apprehend you for the state. The bail bonds entity will then sue you for breech of contract in addition to the additional charges from the state. All in all, very bad for you.