Specter is how Americans spell 'spectre'. British and most other writers use 'spectre'; the meaning is identical, only the last two letters swap. It has two lives. The first is literal: a ghost — but a particular flavour of ghost, one that appears silently and frightens you. It is a literary word. In a novel or a poem you meet a specter drifting through a dark house; in normal conversation people say 'ghost'. If you say 'I saw a specter' to a friend, it sounds like you are joking or being dramatic. The second life is the one you will actually read in newspapers, and it is more useful. A specter is a frightening possibility that hangs over a situation — something bad that has not happened but keeps threatening to. 'The specter of unemployment', 'the specter of war', 'raise the specter of another crisis'. The image is that the bad thing is haunting people, like a ghost, even though it is not real yet. Two phrases carry most of this use: 'the specter of X' and 'raise the specter of X'. This is formal, written English — journalism, essays, speeches. Knowing this second use is worth much more to a student than knowing the ghost meaning.