There's an important facet to instanceof that does not seem to be covered in any of the comments thus far: inheritance. A variable being evaluated by use of instanceof could return true for multiple "types" due to prototypal inheritance.

For example, let's define a type and a subtype:

function Foo(){ //a Foo constructor
    //assign some props
    return this;
}

function SubFoo(){ //a SubFoo constructor
    Foo.call( this ); //inherit static props
    //assign some new props
    return this;
}

SubFoo.prototype = Object.create(Foo.prototype); // Inherit prototype
SubFoo.prototype.constructor = SubFoo;

Now that we have a couple of "classes" lets make some instances, and find out what they're instances of:

var 
    foo = new Foo()
,   subfoo = new SubFoo()
;

alert( 
    "Q: Is foo an instance of Foo? "
+   "A: " + ( foo instanceof Foo ) 
); // -> true

alert( 
    "Q: Is foo an instance of SubFoo? " 
+   "A: " + ( foo instanceof SubFoo ) 
); // -> false

alert( 
    "Q: Is subfoo an instance of Foo? "
+   "A: " + ( subfoo instanceof Foo ) 
); // -> true

alert( 
    "Q: Is subfoo an instance of SubFoo? "
+   "A: " + ( subfoo instanceof SubFoo ) 
); // -> true

alert( 
    "Q: Is subfoo an instance of Object? "
+   "A: " + ( subfoo instanceof Object ) 
); // -> true

See that last line? All "new" calls to a function return an object that inherits from Object. This holds true even when using object creation shorthand:

alert( 
    "Q: Is {} an instance of Object? "
+   "A: " + ( {} instanceof Object ) 
); // -> true

And what about the "class" definitions themselves? What are they instances of?

alert( 
    "Q: Is Foo an instance of Object? "
+   "A:" + ( Foo instanceof Object) 
); // -> true

alert( 
    "Q: Is Foo an instance of Function? "
+   "A:" + ( Foo instanceof Function) 
); // -> true

I feel that understanding that any object can be an instance of MULTIPLE types is important, since you my (incorrectly) assume that you could differentiate between, say and object and a function by use of instanceof. As this last example clearly shows a function is an object.

This is also important if you are using any inheritance patterns and want to confirm the progeny of an object by methods other than duck-typing.

Hope that helps anyone exploring instanceof.

The other answers here are correct, but they don't get into how instanceof actually works, which may be of interest to some language lawyers out there.

Every object in JavaScript has a prototype, accessible through the __proto__ property. Functions also have a prototype property, which is the initial __proto__ for any objects created by them. When a function is created, it is given a unique object for prototype. The instanceof operator uses this uniqueness to give you an answer. Here's what instanceof might look like if you wrote it as a function.

function instance_of(V, F) {
  var O = F.prototype;
  V = V.__proto__;
  while (true) {
    if (V === null)
      return false;
    if (O === V)
      return true;
    V = V.__proto__;
  }
}

This is basically paraphrasing ECMA-262 edition 5.1 (also known as ES5), section 15.3.5.3.

Note that you can reassign any object to a function's prototype property, and you can reassign an object's __proto__ property after it is constructed. This will give you some interesting results:

function F() { }
function G() { }
var p = {};
F.prototype = p;
G.prototype = p;
var f = new F();
var g = new G();

f instanceof F;   // returns true
f instanceof G;   // returns true
g instanceof F;   // returns true
g instanceof G;   // returns true

F.prototype = {};
f instanceof F;   // returns false
g.__proto__ = {};
g instanceof G;   // returns false

I think it's worth noting that instanceof is defined by the use of the "new" keyword when declaring the object. In the example from JonH;

var color1 = new String("green");
color1 instanceof String; // returns true
var color2 = "coral";
color2 instanceof String; // returns false (color2 is not a String object)

What he didn't mention is this;

var color1 = String("green");
color1 instanceof String; // returns false

Specifying "new" actually copied the end state of the String constructor function into the color1 var, rather than just setting it to the return value. I think this better shows what the new keyword does;

function Test(name){
    this.test = function(){
        return 'This will only work through the "new" keyword.';
    }
    return name;
}

var test = new Test('test');
test.test(); // returns 'This will only work through the "new" keyword.'
test // returns the instance object of the Test() function.

var test = Test('test');
test.test(); // throws TypeError: Object #<Test> has no method 'test'
test // returns 'test'

Using "new" assigns the value of "this" inside the function to the declared var, while not using it assigns the return value instead.

And you can use it for error handling and debugging, like this:

try{
    somefunction();
} 
catch(error){
    if (error instanceof TypeError) {
        // Handle type Error
    } else if (error instanceof ReferenceError) {
        // Handle ReferenceError
    } else {
        // Handle all other error types
    }
}

What is it?

Javascript is a prototypal language which means it uses prototypes for 'inheritance'. the instanceof operator tests if a constructor function's prototype propertype is present in the __proto__ chain of an object. This means that it will do the following (assuming that testObj is a function object):

obj instanceof testObj;

1. Check if prototype of the object is equal to the prototype of the constructor: obj.__proto__ === testObj.prototype >> if this is true instanceof will return true.
2. Will climb up the prototype chain. For example: obj.__proto__.__proto__ === testObj.prototype >> if this is true instanceof will return true.
3. Will repeat step 2 until the full prototype of object is inspected. If nowhere on the prototype chain of the object is matched with testObj.prototype then instanceof operator will return false.

Example:

function Person(name) {
  this.name = name;
}
var me = new Person('Willem');

console.log(me instanceof Person); // true
// because:  me.__proto__ === Person.prototype  // evaluates true

console.log(me instanceof Object); // true
// because:  me.__proto__.__proto__ === Object.prototype  // evaluates true

console.log(me instanceof Array);  // false
// because: Array is nowhere on the prototype chain

What problems does it solve?

It solved the problem of conveniently checking if an object derives from a certain prototype. For example, when a function recieves an object which can have various prototypes. Then, before using methods from the prototype chain, we can use the instanceof operator to check whether the these methods are on the object.