What exactly is our mission in the Libya War?

The publicly stated reason is to prtect Libyan civilians. The actual reason is to engineer the overthow of Gaddafi or engineer the partition of Libya, so that the rebels end up controlling the eastern side of the country, which of course is where the oil is. If it had even the slightest thing to do with saving civilian lives, NATO would be bombing Yemen, Bahrain, Saudi, Israel etc; all nations with a proven record of mass murder of civilian protestors in very recent times. Except they are all 'allies', so they get a free pass. Gaddafi, on the other hand, wanted to nationalise Libya's oil (the largest reserves in Africa), which was what put him in the crosshairs of Britain, France, and the USA.