At the end of Return of the Jedi, you see many planets celebrating the death of Palpatine, Vader, and the destruction of Death Star II.
Does the death of the Emperor mean the end of the Empire? I would assume Vader would be second in command to take the Emperor's place, and someone third in command to take Vader's place if need be.
Or perhaps, there would be Senate members or some type of political group who would decide on the Empire's restructure?
In the years after the victory at Endor the Rebel Alliance molds itself into the New Republic.
This era of Star Wars history is covered in the Expanded Universe, specifically the books under the "New Republic" banner, which cover the first twenty years after Endor.
Wookieepedia has a list.
Essentially, there are still battles to be fought, such as the taking of Coruscant in the second book of the X-Wing series, Wedges' Gamble. The closest person to the 3rd in command who takes control of the Empire after the deaths of Vader and Palpatine would be Ysanne Isard:
Ysanne Isard was a female Human who served as the Director of Imperial Intelligence for several years during the Galactic Civil War, and ruled the Galactic Empire herself for a time after the Emperor's death in 4 ABY.
Other characters are shown leading the Empire (such as Paltr Carvin), but they don't last long.
After that in the first modern EU books (released in the early 90s) the Thrawn Trilogy by Timothy Zahn (set 9 years after Episode IV), Grand Admiral Thrawn takes control of the remnant of the Empire:
(Also probably a good place to start with the EU)
Speaking strictly G-Canon/Non-EU:
Just because Papaltine and Vader have been killed, it doesn't mean the end of the rebellion
Remember Tarkin's delivery of news in ANH:
TARKIN
The Imperial Senate will no longer be of any concern to us. I've just received word that the Emperor has dissolved the council permanently. The last remnants of the Old Republic have been swept away.
TAGGE
That's impossible! How will the Emperor maintain control without the bureaucracy?
TARKIN
The regional governors now have direct control over territories. Fear will keep the local systems in line. Fear of this battle station.
Yes, both Death Stars have been destroyed and the head of the Empire is gone - but regional governors and military still exist and maintain a hold in each system.
If something were to happen to the Federal government in the US, there would still be the state, county and city governments which control would fall back to.
After the war the Rebel Alliance, now just called the Galactic Alliance, and whats left of the empire, commonly called The Imperial Remnant, have a ceasefire and more or less live in peace.
