What is a callback function?

The Callback page on Wikipedia explains it very well:

In computer programming, a callback is a reference to executable code, or a piece of executable code, that is passed as an argument to other code. This allows a lower-level software layer to call a subroutine (or function) defined in a higher-level layer.

A callback function is one that should be called when a certain condition is met. Instead of being called immediately, the callback function is called at a certain point in the future.

Typically it is used when a task is being started that will finish asynchronously (ie will finish some time after the calling function has returned).

For example, a function to request a webpage might require its caller to provide a callback function that will be called when the webpage has finished downloading.

A callback function is a function you specify to an existing function/method, to be invoked when an action is completed, requires additional processing, etc.

In Javascript, or more specifically jQuery, for example, you can specify a callback argument to be called when an animation has finished.

In PHP, the preg_replace_callback() function allows you to provide a function that will be called when the regular expression is matched, passing the string(s) matched as arguments.

Call After would be a better name than the stupid name, callback. When or if condition gets met within a function, call another function, the Call After function, the one received as argument.

Rather than hard-code an inner function within a function, one writes a function to accept an already-written Call After function as argument. The Call After might get called based on state changes detected by code in the function receiving the argument.

A callback is an idea of passing a function as a parameter to another function and have this one invoked once the process has completed.

If you get the concept of callback through awesome answers above, I recommend you should learn the background of its idea.

"What made them(Computer-Scientists) develop callback?" You might learn a problem, which is blocking.(especially blocking UI) And callback is not the only solution to it. There are a lot of other solutions(ex: Thread, Futures, Promises...).

A callback function is a function you pass (as a reference or a pointer) to a certain function or object. This function or object will call this function back any time later, possibly multiple times, for any kind of purpose :

• notifying the end of a task
• requesting comparison between two item (like in c qsort())
• reporting progress of a process
• notifying events
• delegating the instanciation of an object
• delegating the painting of an area

...

So describing a callback as a function being called at the end of another function or task is overly simplifying (even if it's a common use case).

One important usage area is that you register one of your function as a handle (i.e. a callback) and then send a message / call some function to do some work or processing. Now after the processing is done, the called function would call our registered function (i.e. now call back is done), thus indicating us processing is done.
This wikipedia link explains quite well graphically.

Callback Function A function which passed to another function as an argument.

function test_function(){       
 alert("Hello world");  
} 

setTimeout(test_function, 2000);

Note: In above example test_function used as an argument for setTimeout function.