An appeal is the formal process of asking for a change in the court or jury decision. Those who wish to make an appeal of their conviction should seek experienced legal help for the best chance of having the appeal considered. However, there is a very short period of time after conviction within which to appeal.
Similar to an arrest warrant, a bench warrant against a person means that they can be arrested on the spot if found by law enforcement. Most commonly, a bench warrant is issued against those who failed to show up in court, or is identified as a subpoened witness who refuses to appear at trial.