Roggvir is a Solitude citizen who worked as a city guard. After Ulfric Stormcloak murdered High King Torygg, he tried to escape the city, and Roggvir opened the gate for him. He claims that he did it because Ulfric bested the High King in fair combat, despite the fact that almost everyone else considers him a traitor.
When the Dragonborn first enters the city of Solitude from its main gate, the execution of Roggvir is taking place. He denies being a traitor while the city’s captain orders the headsman to execute him by axe.
During the execution, it is forbidden to access the platform where the execution is being performed. Doing so will make the guards and the captain hostile, and they will try to kill Roggvir as well.
After the execution has taken place, an Amulet of Talos can be looted from Roggvir’s corpse, which can be used in a quest related to his death.
If the Dragonborn succeeds in saving Roggvir from execution (which is very difficult, as once the Dragonborn attacks any of the people on the stand or the crowd, the captain attacks Roggvir, killing him in one strike), Roggvir doesn’t seem to care. No dialogue options open, and he will treat the Dragonborn like a normal person would. In other words, just let Roggvir die, as the script depends on it.