Prison Architect will have you face many different kinds of prisoners. One such kind of prisoner is Death Row inmates. These are prisoners that are ready to face the death sentence, provided their appeals don’t work.
In order to execute prisoners, they must be a Death Row inmate. You can only call these rare prisoners once you’ve established a Death Row sector in your prison. To do this, build a section of cells away from others (and preferably close to where you will build the execution room) and establish it as the Death Row Sector in your Security Sectors section. Don’t forget to put a guard on duty there, as the inmates that come to it will be very, very mean.
Next, build an execution room complete with an execution chair. You’re almost set, the only thing you should do after you’ve received your Death Row inmate is wait and allow appeals to happen. These are scheduled as a Program and happen in the Parole Hearing room that you must build. If you click on the inmate, you’ll find out what percentage of conviction they are. It’s wise to wait until they’ve failed enough appeals before you go through with their execution, otherwise you will face a lawsuit. The more appeals they fail, the higher the percentage goes up. They can, of course, be let go if they are found not guilty.
Finally, once your prisoner is definitely not going to be found innocent, you can proceed with the execution plans from the inmate character menu (clicking on them). Go through each of the steps until the final one which will have your prisoner executed in Prison Architect.