What is the significance of each of their patronuses and how do they work?

I am going to quote from Writings by J.K. Rowling on Pottermore.com because there is nothing more reliable source than J.K. Rowling herself.

No reliable system for predicting the form of an individual’s Patronus has ever been found, although the great eighteenth-century researcher of Charms, Professor Catullus Spangle, set forth certain principles that are widely accepted as true.

The Patronus, asserted Spangle, represents that which is hidden, unknown but necessary within the personality.

‘For it is evident,’ he writes, in his masterwork ‘Charms of Defence and Deterrence’, ‘… that a human confronted with inhuman evil, such as the Dementor, must draw upon resources he or she may never have needed, and the Patronus is the awakened secret self that lies dormant until needed, but which must now be brought to light...’

Here, says Spangle, is the explanation for the appearance of Patronuses in forms that their casters might not expect, for which they have never felt a particular affinity, or (in rare cases) even recognize. Spangle is interesting on the subject of those unusual witches and wizards who produce a Patronus that takes the form of their favorite animal.

‘It is my firm belief that such a Patronus is an indicator of obsession or eccentricity. Here is a wizard who may not be able to hide their essential self in common life, who may, indeed, parade tendencies that others might prefer to conceal. Whatever the form of their Patronus, you would be well-advised to show respect, and occasionally caution, towards a witch or wizard who produces the Patronus of their choice.’

A patronus takes the shape of whatever animal the said person has a special relationship with. To create a competent patronus, one must think of a happy memory and say an incantation. Happiness repels and repulses dementors, being foul things that thrive off of sadness and despair.

Each patronus links to someone's personal life. Harry's patronus is a stag because his father was a stag animagus. Snape's patronus is a doe because he loved Lily and her patronus was a doe. you patronus can also link to your personality. In the books, Tonks patronus changed because she was no longer her normal self after she was sad about Sirius.