std::function and std::bind: what are they, and when should they be used?

std::bind is for partial function application.

That is, suppose you have a function object f which takes 3 arguments:

f(a,b,c);

You want a new function object which only takes two arguments, defined as:

g(a,b) := f(a, 4, b);

g is a "partial application" of the function f: the middle argument has already been specified, and there are two left to go.

You can use std::bind to get g:

auto g = bind(f, _1, 4, _2);

This is more concise than actually writing a functor class to do it.

There are further examples in the article you link to. You generally use it when you need to pass a functor to some algorithm. You have a function or functor that almost does the job you want, but is more configurable (i.e. has more parameters) than the algorithm uses. So you bind arguments to some of the parameters, and leave the rest for the algorithm to fill in:

// raise every value in vec to the power of 7
std::transform(vec.begin(), vec.end(), some_output, std::bind(std::pow, _1, 7));

Here, pow takes two parameters and can raise to any power, but all we care about is raising to the power of 7.

As an occasional use that isn't partial function application, bind can also re-order the arguments to a function:

auto memcpy_with_the_parameters_in_the_right_flipping_order = bind(memcpy, _2, _1, _3);

I don't recommend using it just because you don't like the API, but it has potential practical uses for example because:

not2(bind(less<T>, _2, _1));

is a less-than-or-equal function (assuming a total order, blah blah). This example normally isn't necessary since there already is a std::less_equal (it uses the <= operator rather than <, so if they aren't consistent then you might need this, and you might also need to visit the author of the class with a cluestick). It's the sort of transformation that comes up if you're using a functional style of programming, though.

One of the main use of std::function and std::bind is as more generelized function pointers. You can use it to implement callback mechanism. One of the popular scenario is that you have some function that is going to take long time to execute but you don't want to wait for it to return then you can run that function on separate thread and give it a function pointer that it will callback after it completes.

Here's a sample code for how to use this:

class MyClass {
private:
    //just shorthand to avoid long typing
    typedef std::function<void (float result)> TCallback;

    //this function takes long time
    void longRunningFunction(TCallback callback)
    {
        //do some long running task
        //...
        //callback to return result
        callback(result);
    }

    //this function gets called by longRunningFunction after its done
    void afterCompleteCallback(float result)
    {
        std::cout << result;
    }

public:
    int longRunningFunctionAsync()
    {
        //create callback - this equivalent of safe function pointer
        auto callback = std::bind(&MyClass::afterCompleteCallback, 
            this, std::placeholders::_1);

        //normally you want to start below function on seprate thread, 
        //but for illustration we will just do simple call
        longRunningFunction(callback);
    }
};

std::bind was voted into library after proposal to include boost bind, primarily it is partial function specialization where-in you can fix few parameters and change others on fly. Now this is library way of doing lambdas in C++. As answered by Steve Jessop

Now that C++11 supports lambda functions I don't feel any temptation to use std::bind anymore. I would rather use currying (partial specialization) with language feature than library feature.

std::function objects are polymorphic functions. The basic idea is to be able to refer to all the callable objects interchangeably.

I would point you to these two links for further details:

Lambda functions in C++11: http://www.nullptr.me/2011/10/12/c11-lambda-having-fun-with-brackets/#.UJmXu8XA9Z8

Callable entity in C++: http://www.nullptr.me/2011/05/31/callable-entity/#.UJmXuMXA9Z8