Definition
A definite description is a phrase of the form the so-and-so — the present king of France, the smallest prime number, the author of Waverley. Unlike a proper name, which simply labels an object, a definite description claims to single out exactly one thing by means of a property the description specifies. Its grammatical job is to refer; its logical job is to refer uniquely.
The trouble is that some descriptions do this work and some do not. The present king of France picks out nobody, because there is no king of France. Yet a sentence like 'the present king of France is bald' is a perfectly grammatical English sentence. What, then, does it mean?
Why it matters
How it works
Bertrand Russell's celebrated theory of descriptions dissolves the puzzle by analysing the description away. 'The so-and-so is F' is treated as a compound claim: there exists something that is a so-and-so; at most one thing is a so-and-so; and that thing is F. On this reading 'the present king of France is bald' makes a definite assertion — and because the first conjunct is false, the whole sentence is simply false. No mysterious non-existent object is needed.
This move shows that a phrase can look like a referring term while functioning logically as a bundle of quantified claims. Surface grammar, Russell argued, is a poor guide to logical form.