Controller

Controller: A visualforce controller is a set of instructions that specify what happens when a user interacts with the components specified  in associated visualforce markup, such as when a user clicks a button or link.

Controller also provide access to the data that should be displayed in a page, and can modify component behavior

StandardController: A standardcontroller consist of the same functionality and logic that is used for a standard salesforce page. for eample , if you use the standar account controller, clicking a save button in a visualforce page results in the same behaviour as clicking save on a Standard Account detail page.

<apex:page standardController="Account" >

Custom Controller: A custom Controller is a class written in Apex that implements all of a page's logic. If we use a custom controller, we can define new navigation elements or behaviour, but we must also reimplement any functionality that was already provided in a standard controller.

<apex:page controller="apex Class Name" >

ControllerExtensions: A controller extension is a class written in apex that adds to or override behavior in a standard or custom controller. Extensions allow us to leverage the functionality of another controller while adding our own custom logic 

<apex:page StandardController="Account" extension="Apex Class Name" >