Being an independent voter means that you cross all party boundaries and vote for the person you feel is most qualified. It means that you are not a member of a political party and can vote for a Republican, Democrat, Libertarian, Green Party, Socialist Party, and etc. candidate if you feel they are the best choice.
So an independent voter can choose a Democratic presidential candidate, a Republican senate candidate, a Libertarian congressional candidate, and a Green party judge candidate in a single ballot if they so choose. This is called voting a "split ticket".
You can not tell from their independence whether a voter is conservative or liberal on social or moral issues, but only by their actual vote. As for what they believe in, it varies from voter to voter. There is no preset ideology that comes from being an independent voter. Unlike members of political parties, independent voters think for themselves, and can be conservative on social issues and liberal on financial issues, or vice versa, or both, or neither.
Generally independent voters see the world in shades of gray, not black and white.