Originating from the military, "mission accomplished" simply means that a mission's goals have been successfully fulfilled.
The trick is defining the mission properly. Technically, a mission is a single complex task within larger operations, battles, and wars. Idiomatically, politicians often use the word mission to refer to any major sustained effort. These two meanings can come into conflict.
Example: In a famous speech, former U.S. President George W. Bush declared major combat operations in Iraq to be over, speaking from an aircraft carrier with a huge "Mission Accomplished" banner visible to the news media. "Minor" combat operations would claim the lives of many Americans and a great many Iraqis in years to come. Here, the literal mission may have been accomplished, but the figurative mission was far from finished, and far from success.