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How to make a window come up on button click?

So basically my goal is to code my own ReCaptcha. To make that possible, I'd like a window to come up once I clicked a button to send a form or something like this. In best case it looks like this, but with my own content inside. Is it possible to do this with just one js-script, or with adding a few divs?

I think you can use jquery modal.
check the following link https://jquerymodal.com/
official docs can be found at
https://jqueryui.com/dialog/

How to send event on plugin?

I am using the leanModal plugin and I like to add some code, in order to meet my needs.
What I did, is to use it as a modal dialog, and I have two buttons. The one button closing the modal dialog, and it is fine. The second button is responsible to use the data inserted into the dialog, and then to close the dialog.
What I am looking now, is when I click the second button, first to validate the data, and then to close the dialog, but this functionality does not exists in the plugin. So, how can I add something like the following:
$('selector').leanModal('close');
Is there a way ?
Note : I hope that my description is ok.

Custom Confirm Form by jQuery

Now I use return confirm(Message); when need confirmation in a form. that cause displaye a form with two button OK and Cancel. Is there any way to change the buttons text in displayed form? I need to change buttons to another language? what is your suggestion?

Since you tagged jquery, jQuery UI Modal Confirm is definitely worth a look
Working fiddle: http://jsfiddle.net/naveen/GR2UT/

You can't change the button text on a native confirm box. You can program your own modal dialog, which would give you control to what's in the buttons. In this jsfiddle you'll find a plain javascript example. Or use JQuery, like naveen suggested.

Modal Windows using jQuery

I'm working on a web app where in I need to add some values inputted by the user.
When the user clicks on the Add button he sees a form and these values then show up on the page. Now to implement this I can do two things
Use a modal window
Make a form inside the page itself(in a div) and toggle its visibility by the Add button.
If I go with the former solution is it necessary to use Ajax or I can add elements on the main page directly itself? Are there any jQuery plugins to accomplish the same?

As Diodeus mentioned, you don't need Ajax.
In answer to your other question about the plugins, have a look at jQuery UI Dialog for the dialog. Generating the form is pretty trivial. There's no need for a plugin there

You don't need to use Ajax to accomplish this. The difference in the two methods is simply whether you use an inline block of code that is hidden and displayed later, or whether you use an absolutely-positioned block of code doing exactly the same thing.
In most cases there is a single form that envelops the entire page. The rest is a matter of a CSS and positioning.

Looking for particular alert box behavior

I'm interested in alert boxes such as,
http://www.visual-blast.com/javascript/nice-alert-box-with-jquery-impromptu/
and
http://woork.blogspot.com/2008/08/design-stunning-alert-box-using.html
but with a different functionality, the user should be able to interact with the page even with the alert box present, so it needs to be movable and not obscure the underlying page, as in this example,
http://www.open-lab.com/mb.ideas/index.html
Alert box should be generated automatically on page view plus the title and message of the alert box should change with each page refresh from expandable library.
Is there anything out there like this?
Thanks

Sounds like the jQuery UI dialog widget might do what you need?
http://jqueryui.com/demos/dialog/

Here is an example with mootools. You easily can extend any element on the page to be draggable. You may use this plugin as a sample of alerts and you'll just need to remove some functionality and add make it draggable.

There are few custom dialogs in moo.rd. One of them is the Custom.Alert:
http://www.moord.it/examples/custom_alert