Murder in any state is a capital offense thus it is called capital murder. Murder is a felony in all states. A felony by definition is any crime that is punishable by a prison sentence of more than one year. The person charged with this crime is called the defendant, the prosecution is the entity trying to convict the defendant. In capital cases like murder and in criminal proceedings in general the prosecuting body is the state. In this case the prosecution is the State of Texas.
Before trial, the prosecution and defense will undergo a period of "discovery". Essentially, they show each other what they think the facts of the case to be so as to allow time to prepare for the case. In criminal proceedings, this discovery process can be months or even years. Eventually the case will be put on the court's calendar(a trial date will be set).
At trial, jury selection(voir dire) will take place. After the jury has been seated, and after instructions are given by the judge, the trial will begin. Opening statements are made by the defense and by the prosecution. Now each side has the opportunity to call witnesses and present evidence as to why they think the defendant is guilty of the crime, or not guilty.
After all witnesses have testified and all the evidence has been examined, the case is sent to the jury for deliberations. The jury makes a decision and then comes back to render a verdict.
If the defendant is found guilty, he is immediately taken to jail. If the defendant is found innocent he is free to exit and carry on his/her life. If a guilty verdict is rendered in a capital case, an appeal is automatic. These appeals may take many years to play out.