What are Custom Controllers , Custom List Controller and Controller Extensions?

A custom controller is an Apex class that implements all of the logic for a page without leveraging a standard controller. Use custom controllers when you want your Visualforce page to run entirely in system mode, which does not enforce the permissions and field-level security of the current user.

Example

<apex:page controller="myController" tabStyle="Account">
    <apex:form>
        <apex:pageBlock title="Congratulations {!$User.FirstName}">
            You belong to Account Name: <apex:inputField value="{!account.name}"/>
            <apex:commandButton action="{!save}" value="save"/>
        </apex:pageBlock>
    </apex:form>

</apex:page>

public class MyController {

    private final Account account;

    public MyController() {
        account = [SELECT Id, Name, Site FROM Account 
                   WHERE Id = :ApexPages.currentPage().getParameters().get('id')];
    }

    public Account getAccount() {
        return account;
    }

    public PageReference save() {
        update account;
        return null;
    }

}

A controller extension is an Apex class that extends the functionality of a standard or custom controller. Use controller extensions when:

You want to leverage the built-in functionality of a standard controller but override one or more actions, such as edit, view, save, or delete.

You want to add new actions.

You want to build a Visualforce page that respects user permissions. Although a controller extension class executes in system mode, if a controller extension extends a standard controller, the logic from the standard controller does not execute in system mode. Instead, it executes in user mode, in which permissions, field-level security, and sharing rules of the current user apply.

<apex:page standardController="Account" extensions="myControllerExtension">
    {!greeting} <p/>
    <apex:form>
        <apex:inputField value="{!account.name}"/> <p/>
        <apex:commandButton value="Save" action="{!save}"/>
    </apex:form>

</apex:page>

public class myControllerExtension {

    private final Account acct;
    
    // The extension constructor initializes the private member
    // variable acct by using the getRecord method from the standard
    // controller.
    public myControllerExtension(ApexPages.StandardController stdController) {
        this.acct = (Account)stdController.getRecord();
    }

    public String getGreeting() {
        return 'Hello ' + acct.name + ' (' + acct.id + ')';
    }

}

Custom List Controller

A custom list controller is similar to a standard list controller. Custom list controllers can implement Apex logic that you define to show or act on a set of records.

<apex:page controller="opportunityList2Con">
    <apex:pageBlock>
        <apex:pageBlockTable value="{!opportunities}" var="o">
            <apex:column value="{!o.Name}"/>
            <apex:column value="{!o.CloseDate}"/>
        </apex:pageBlockTable>
    </apex:pageBlock>

</apex:page>

public class opportunityList2Con {
    // ApexPages.StandardSetController must be instantiated
    // for standard list controllers
    public ApexPages.StandardSetController setCon {
        get {
            if(setCon == null) {
                setCon = new ApexPages.StandardSetController(Database.getQueryLocator(
                    [SELECT Name, CloseDate FROM Opportunity]));
            }
            return setCon;
        }
        set;
    }

    // Initialize setCon and return a list of records
    public List<Opportunity> getOpportunities() {
        return (List<Opportunity>) setCon.getRecords();
    }

}

Note -

Custom controllers and controller extension classes execute in system mode, so they ignore user permissions and field-level security. However, you can choose whether they respect a user's organization-wide defaults, role hierarchy, and sharing rules by using the with sharing keywords in the class definition. For information, see “Using the with sharing, without sharing, and inherited sharing Keywords” in the Apex Developer Guide.