Dispute solving topic - courts hierarcyWatch
Understand that an indictable offence goes to the Crown Court and a summary offence goes to a Magistrates'. The key difference between the two courts is that the Magistrates' is limited in its sentencing powers to 6 months for a single offence and 12 months for a collection of summary offences. As you can infer, indictable offences are the more serious offences and cover offences like murder and rape; and summary offences are crimes like common assault. An offence which exists between these two is an offence triable either way—the most common offence triable either way is theft. The Magistrates' will come to the determination of whether or not the court has sufficient sentencing powers to deal with the offence by reviewing the facts of the case*. If they determine they do not have sufficient sentencing powers, they will send it to be heard before a jury and a judge at a Crown Court. If they decide they can deal with the offence themselves, they still must give the defendant the choice of whether he wants to be tried summarily or by indictment.
I can't really answer based on appeals because I'll be here all day. But basically, appeals are given permission by the court against which you are appealing. The case will then be heard by a judge one rung up the hierarchy. County court to High Court, and High Court to Court of Appeal. Then High Court to Court of Appeal, and Court of Appeal to Supreme Court. Criminal cases start at Magistrates'. Appeals from Magistrates' goes to the Crown Court, appeals from Crown Court goes to High Court (Queen's Bench Division), cases from HC QBD go to the Court of Appeal, and cases from the Court of Appeal go to the Crown Court. Cases can be 'leapfrogged' from the High Court to the Supreme Court where there is a matter of public importance at hand and other courts cannot deal with the issue due to being bound by precedent. The Supreme Court can obviously overturn precedent where it deems appropriate to do so. This again is very basic and you need to explore depending on the depth of the essay you are writing. There are many ways which a case from a Magistrates' can be overturned by QBD, for example.
*There are several exceptions which you perhaps need to explore depending on the depth required by the essay. Magistrates' also consider the complexity of the offence, as 'professional' judges may be better suited to understand the case and have more time to deal with it.