How to call a controller method from jQuery/javascript

jQuery is JavaScript. Just call the function normally. You can call an actionFunction, RemoteAction, and so on.

As a trivial example, here's a simple incremental controller. Every time you call increment, the value increases by 1:

public class counterController {
    public Integer counter { get; set; }
    public counterController() {
        counter = 0;
    }
    public void increment() {
        counter++;
    }
}

In the page, I'll use jQuery to bind to the keypress event, which in turn will call the controller's "increment" function:

<apex:page controller="counterController">
    <script 
    src="https://code.jquery.com/jquery-3.1.0.slim.min.js"  
    integrity="sha256-cRpWjoSOw5KcyIOaZNo4i6fZ9tKPhYYb6i5T9RSVJG8=" 
    crossorigin="anonymous">
        </script>
    <script>
    $(function() {
        $(window).on("keypress", function() {
            alert("I'm about to increment the counter.");
            incrementCounter();
        });
    });
    </script>
    <apex:form id="form">
        Press a key to increment the counter.
        <apex:actionFunction name="incrementCounter" action="{!increment}" reRender="form" />
        The counter is now {!counter}.
    </apex:form>
</apex:page>

You can use this type of code with JavaScript Remoting for Apex Controllers

Visualforce

<script type="text/javascript">
 var j$ = jQuery.noConflict(); 
enter j$(document).ready(function(){
j$("#Submit").click(function() 
    {       
      MyController.updateMethod("ABC", function(result, event){
            console.log(result);
        });
    }); 

});
</script>

Controller

public class MyController{
public list <MyObject__c> pub {get;set;}
public MyController(ApexPages.StandardController controller) {
}


@RemoteAction
public static void updateMethod(MyObject__c myObj)
{
    //do some update

    update myObj;
}
}