At the end of Star Wars: Episode III – Revenge of the Sith Bail Organa orders Captain Antilles to wipe C-3PO's memory.
I'm placing these droids in your care. Treat them well. Clean them up. Have the protocol droid's mind wiped.
Why wasn't R2-D2's memory wiped as well?
A memory wipe was a method that was used to selectively or completely erase the memory of a droid. After Cad Bane had kidnapped and ransacked the memory of C-3PO and R2-D2 for information on the Senate Building's layout, he only had the incident wiped from their memories to avoid having any suspicion falling on him. Shortly after the Galactic Empire was formed, Bail Organa had C-3PO's memory wiped clean in order for the droid to not spill any confidential information. When he purchased C-3PO and R2-D2, Owen Lars also wanted the droids' memories wiped after he learned they were looking for Obi-Wan Kenobi.
Memory wipes were an important process, and the lack of one could allow droids to develop new ideas. This happened with the mining droid STALKER-1, who turned into a hunter after years without a memory reset. Memory wipes were far from perfect, however, and both C-3PO and the First Order droid O-MR1 both experienced flashes of erased memories. A macro protocol could be used to wipe the memories[6] of multiple droids simultaneously.
The Galactic Empire did not grant its droids any independence and sought to keep them subservient and docile. Part of this process was frequent memory wipes in order to keep them from getting independent notions. http://starwars.wikia.com/wiki/Memory_wipe
So although this does not explain the mystery of why R2-D2 did NOT have his memory swiped at this time, it does confirm the idea that C-3PO would be a droid that would be prone to giving out pertinent information, given the characters penchant for excessive rambling and storytelling.
The wookieepedia article also makes point that memory or memory-swiping is something slightly thematic to Star Wars, particularly with droids and a few of the ways memory swipes can be used in different ways from espionage tactics with hiding information to themes about slavery and [identity] suppression.
The answer to this question should be fairly obvious to anyone who has watched all the movies.
Star Wars (later re-titled A New Hope, Episode 4) was the first movie released. At that time they couldn't have any idea that this movie would be the blockbuster that it was spawning so many sequels and prequels. Point being it was episode 4! When it came time to make the prequels, the writers had to reverse engineer some plot points in order to explain why the characters behaved a certain way in what is actually a later timeline.
In episode 4, (the original Star Wars movie), when Luke accidentally plays back a portion of the video file of Princess Leia, C-3PO is truly clueless as to who this person is an only knows that she was "a person of some importance" on the ship in which they served on. I think it could also be speculated that C-3PO would be a terrible liar and would not be able to play off his naïveté so genuinely.
R2-D2 is clearly lying through his teeth (figuratively of course), he very well knows the identity of the Princess, has been given a secret mission and tries to hide any information about her from Luke. So it was absolutely imperative when episode 3 finally does get made so many years later that this somehow gets addressed. Wiping C-3PO's memory or at least a portion of it and leaving R2-D2 and his memory intact fills this plot hole quite reasonably.
