A signature dish is a recipe that identifies an individual chef. Ideally it should be unique and allow an informed gastronome to name the chef in a blind tasting. It can be thought of as the culinary equivalent of an artist finding their own style, or an author finding their own voice. In practice a chef's signature dish often changes with time or they may claim several signature dishes.
In a weaker sense, a signature dish may become associated with an individual restaurant, particularly if the chef who created it has since moved on or died. It can also be used to refer to a culinary region, in which case its meaning may be the equivalent of 'national dish'.
At its weakest, in a mediocre restaurant, the term can simply mean 'Chef's Specials' which are in no way unique or even particularly unusual.
Examples of signature dishes are;