What is the difference between a definition and a declaration?

A declaration introduces an identifier and describes its type, be it a type, object, or function. A declaration is what the compiler needs to accept references to that identifier. These are declarations:

extern int bar;extern int g(int, int);double f(int, double); // extern can be omitted for function declarationsclass foo; // no extern allowed for type declarations

A definition actually instantiates/implements this identifier. It's what the linker needs in order to link references to those entities. These are definitions corresponding to the above declarations:

int bar;int g(int lhs, int rhs) {return lhs*rhs;}double f(int i, double d) {return i+d;}class foo {};

A definition can be used in the place of a declaration.

An identifier can be declared as often as you want. Thus, the following is legal in C and C++:

double f(int, double);double f(int, double);extern double f(int, double); // the same as the two aboveextern double f(int, double);

However, it must be defined exactly once. If you forget to define something that's been declared and referenced somewhere, then the linker doesn't know what to link references to and complains about a missing symbols. If you define something more than once, then the linker doesn't know which of the definitions to link references to and complains about duplicated symbols.

Since the debate what is a class declaration vs. a class definition in C++ keeps coming up (in answers and comments to other questions) , I'll paste a quote from the C++ standard here.
At 3.1/2, C++03 says:

A declaration is a definition unless it [...] is a class name declaration [...].

3.1/3 then gives a few examples. Amongst them:

[Example: [...]struct S { int a; int b; }; // defines S, S::a, and S::b [...]struct S; // declares S—end example

To sum it up: The C++ standard considers struct x; to be a declaration and struct x {}; a definition. (In other words, "forward declaration" a misnomer, since there are no other forms of class declarations in C++.)

Thanks to litb (Johannes Schaub) who dug out the actual chapter and verse in one of his answers.

From the C++ standard section 3.1:

A declaration introduces names into a translation unit or redeclares names introduced by previous declarations. A declaration specifies the interpretation and attributes of these names.

The next paragraph states (emphasis mine) that a declaration is a definition unless...

... it declares a function without specifying the function’s body:

void sqrt(double);  // declares sqrt

... it declares a static member within a class definition:

struct X{    int a;         // defines a    static int b;  // declares b};

... it declares a class name:

class Y;

... it contains the extern keyword without an initializer or function body:

extern const int i = 0;  // defines iextern int j;  // declares jextern "C"{    void foo();  // declares foo}

... or is a typedef or using statement.

typedef long LONG_32;  // declares LONG_32using namespace std;   // declares std

Now for the big reason why it's important to understand the difference between a declaration and definition: the One Definition Rule. From section 3.2.1 of the C++ standard:

No translation unit shall contain more than one definition of any variable, function, class type, enumeration type, or template.

Declaration: "Somewhere, there exists a foo."

Definition: "...and here it is!"