All three terms are very similar, but they have different legal definitions. Extortion is a crime that involves obtaining something, usually money, from a person by force or wrongful use of authority or power. With extortion, it is typically someone who has power to cause harm to an innocent person that is forcing them to either give up money or valuables. The victim of extortion is normally innocent of anything themselves. Blackmail is different in that the money or valuable object is not extorted by a threat of direct bodily harm, but rather by the threat of revealing something presumed to be harmful to the victim. The victim of blackmail is typically someone who has either committed a crime they want to stay hidden or they have done something that would be embarrassing to them or cause a loss of job or family. The blackmailer typically knows how much the victim wants to keep the activity secret and therefore agrees to keep it secret for a sum of money or valuable object. Shakedown is a much less formal word and I think may even be slang. Shakedown is more similar to extortion than blackmail.

Extortion (two 't's', not an 's') is more formal and more general than Blackmail. It can be the formal name of a crime, and it can involve just a threat of violence or damage, whereas Blackmail is more a threat to reveal information about someone. I have never used 'shakedown' - I suspect it may be more of an American word. I suspect it refers to more low-end, direct 'stand-over tactics', where some thugs demand money in exchange for the victim not being hurt.