After a DUI arrest, DUI court proceedings will be where guilt or innocence is decided, and where a sentence is given, if guilt is proven. There are a number of individual steps in DUI court proceedings. However, not all of these steps will be a factor in every court case. Additionally, a significant number of DUI cases end in plea bargains or guilty pleas, meaning many do not get all the way through the process.
The first stage of all DUI court proceedings is the arraignment. This is where you will be informed of the charges against you and will be asked how you intend to plead (generally guilty or not guilty). From here, there will be several pretrial stages, some of which will depend on the circumstances of the DUI and where the case is being tried. Generally, these include pretrial conferences, pretrial motions, and possibly a preliminary hearing. Once these are finished, the actual trial begins. Keep in mind that a "failure to appear" doesn't just apply to the trial - missing any of these pre-trial events can still land you in significant trouble.
The trial itself may be either a jury trial or a bench trial (judge trial). In either case, this will be where most evidence is presented, as well as where a verdict will be given in the case of a "not guilty" plea. In the event of a guilty verdict, DUI's tend to be subject to "mandatory minimums" which require a certain level of punishment.