Controllers are attached to your page using one of three mutually exclusive formats, standard controller, standard list/set controller, custom controller.

For a page with a single record, accessing standard data and behavior of your object you would use standard controller:

 

<apex:page standardController="Account"...> ... </apex:page>

 

If you want a page with many records, using standard list view, and list actions you could use standard list controller:
 

<apex:page standardController="Account" recordsetVar="accountsList"...> ... </apex:page>

If you want to use completely custom functionality and don't really need most of the standard object behaviors, custom controller is a good option, but bear in mind, any standard actions you want to use will have to be implemented in your custom class. A page with a custom controller would use the following attribute:

<apex:page controller="MyApexClass"...> ... </apex:page>

But bear in mind, you yourself can instantiate ApexPages.StandardController to use its features and functionality. I often use the pattern below to instantiate standard controller to get access to its methods:
 

public with sharing class MyApexClass {
  private ApexPages.StandardController ctrl;
  public Account acct {get;set;}
  public MyApexClass(){
    try{
      acct = [select Name from Account 
          where Id = : ApexPages.currentPage().getParameters().get('id')];
    } catch (QueryException e) {
      acct = new Account();
    }
      ctrl = new ApexPages.StandardController(acct);
  }
  public PageReference save(){
    //here I can put additional functionality or just leave it as-is.
    return ctrl.save();
  }
}

The above instantiates Standard Controller privately, but then surfaces a public save method that is just a pass-through of the controller's save method. The constructor does the housekeeping of creating the actual record I'm working with.
If I want to add to the functionality of any controller, I can then use an extension. Extensions work with either standard or custom controllers. Standard controller with custom extension, for me, is the most common construct I use for Visualforce that will operate inside my org. I get all the power of the standard controller, but all the flexibility of Apex. The syntax for an extension with a standard controller page would be as follows:
 

<apex:page standardController="account" extensions="MyExtClass"...> ... </apex:page>

The extension class is then required to implement a constructor that accepts the type of the controller. You will then typically want to get a handle to the standard controller instance that is floating around, and the sObject instance that it represents. So for a standard controller page:
 

public with sharing class MyExtClass {
  private ApexPages.StandardController ctrl;
  public Account acct {get;set;}
  public MyExtClass(ApexPages.StandardController controllerParam){
      ctrl = controllerParam;
      acct = (Account) ctrl.getRecord();
  }
  public PageReference customsave(){
    //here I override my standard save functionality before I return the save call.
    return ctrl.save();
  }
}

With a custom controller and extension, you can probably extrapolate from the above, but just to close the loop.
 

<apex:page controller="MyApexClass" extensions="MyExtClass"...> ... </apex:page>

Again, the extension class implements a constructor that accepts the type of the controller, only now it is my custom class. From there, apart from getting a local handle to your custom controller instance, there isn't a pattern, per-se, but the local handle allows you to interact directly with your custom controller, in whatever way necessary, of course. :
 

public with sharing class MyExtClass {
  private MyApexClass ctrl;
  public MyExtClass(MyApexClass controllerParam){
      ctrl = controllerParam;
  }
}

Hope this helps you!

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Thanks and Regards
Sandhya