post data to PHP page in external server and load content from JavaScript in local computer

Here I'll give you a pretty good example of how these things are usually managed.
Still, it's up to you and your programming experience to grasp the meaning of it.

html and js example:

<form action="" id="formId" method="post" accept-charset="utf-8">
    <label for="inputNumber">Input something: </label>
    <input type="number" id="inputNumber" name="count"></input>
</form>
<span id="submit">Submit</span>

<script>
var getPhpResponse = function( data ) {
    console.log("manage php response HERE");
}

$("#submit").click(function(){
    $("#formId").submit();
});

$(document).ready(function () {
        $("#formId").bind("submit", function (event)
        {
            $.ajax({
                async: true,
                data: $("#formId").serialize(),
                success: function(data, textStatus) {
                    getPhpResponse( data )
                },
                type:"POST",
                url:"name/and/location/of/php/file.php"
            });
            return false;
        });
});
</script>

file.php example:

<?php
$x = $_POST['count'];
echo '{"response":"';
for ($i = 0; $i < $x; $i++)
{
    echo $i;
}
echo '"}';

Poxriptum:

1. There should be further input validation, one can't trust the type="number" just yet.
2. That the submit button is a span instead of an input is a personal choice that makes difference just for styling purposes.
3. You should read up on AJAX and JSON.
4. Consider using a PHP framework, such as CakePHP; it may serve you well.
5. This answer assumes you have access to the server. If you don't, then you should be reading the API documentation instead of asking questions on SO without even detailing which API you are talking about.

Edit:

Here is the $less version.

<form action="" id="formId" method="post" accept-charset="utf-8">
    <label for="inputNumber">Input something: </label>
    <input type="number" id="inputNumber" name="count"></input>
</form>
<span id="submit">Submit</span>

<script>
document.getElementById("submit").onclick = function () {
    var url = 'name/and/location/of/php/file.php';
    var userInput = encodeURIComponent(document.getElementById("inputNumber").value);
    var data = "count=" + userInput;
    makeRequest( data, url );
};

var getPhpResponse = function( data ) {
    console.log("manage php response HERE");
    console.log(data);
    parsed = JSON.parse(data);
    console.log(parsed);
}

var xhr = new XMLHttpRequest();

var makeRequest = function( data, url ) {
    xhr.open('POST', url, true);
    xhr.setRequestHeader('Content-Type', 'application/x-www-form-urlencoded; charset=UTF-8');
    xhr.send(data);
};

var xhr = new XMLHttpRequest();
xhr.onreadystatechange = function()
{
    if ( xhr.readyState == 4 )
    {
        if ( xhr.status == 200 || window.location.href.indexOf("http") == -1 )
        {
            getPhpResponse(xhr.responseText);
        }
        else
        {
            console.log("Manage error here");
        }
    }
}

</script>