Meaning
A secret source of shame, potentially ruinous if exposed, which a person or family makes efforts to conceal.

Origin
The phrase 'a skeleton in the closet' was coined in England in the 19th century. Since then the word closet has become used primarily in England to mean 'water closet', i.e. lavatory - a possible hiding place for a skeleton I suppose, but not one with much potential. The English now usually use 'a skeleton in the cupboard', with 'skeleton in the closet' more common in the USA.

'A skeleton in the closet' undoubtedly originated as an allusion to an apparently irreproachable person or family having a guilty secret waiting to be uncovered. The close-at-hand domestic imagery of a closet or cupboard gives a sense of the ever-present risk of discovery. What isn't clear is whether the origin of the phrase lies in fiction or with real life, so to speak, skeletons.
The phrase was first used in the early 1800s. The first reference I can find in print is a figurative one in a piece by William Hendry Stowell, in the UK monthly periodical The Eclectic Review, 1816. The 'skeleton' in this case was the desire to keep a hereditary disease secret:

Two great sources of distress are the danger of contagion and the apprehension of hereditary diseases. The dread of being the cause of misery to posterity has prevailed over men to conceal the skeleton in the closet...

From: http://www.phrases.org.uk