The two controllers that are available to you are:

• StandardController
• StandardSetController

StandardController

From the docs: 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. Controllers also provide access to the data that should be displayed in a page, and can modify component behavior.

You can find more information here, but basically a StandardController is for ONE record, i.e. if you'd like to create a new Visualforce page for a single record, you'd use a Standard Controller in your Apex. For example:

public with sharing class MyController {
    private ApexPages.StandardController controller;

    public MyController(ApexPages.StandardController controller) {
        this.controller = controller;
    }   

    // maybe this gets called from a button within your VF page
    public void doSomethingWithSingleRecord() {
        SObject record = controller.getRecord();

        record.put('Status__c', true);
        update record;
    }

}

StandardSetController

From the docs: Standard list controllers allow you to create Visualforce pages that can display or act on a set of records. Examples of existing Salesforce pages that work with a set of records include list pages, related lists, and mass action pages.

You can find more information here, but basically Set (list) controllers are for MULTIPLE (or a list of) records, i.e. if you'd like to create a new Visualforce page for a List of records (or even from a selection of records on a List view), you'd use a Standard Set Controller in your Apex. For example:

public with sharing class MyController {
    private ApexPages.StandardSetController setController;

    public MyController(ApexPages.StandardSetController setController) {
        this.setController = setController;
    }

    //maybe this gets called from a button on your list view
    public void doSomethingWithMultipleRecords() {
        List<SObject> records = setController.getSelected();

        for(SObject record : records) {
            record.put('Status__c', true);
        }

        update records;
    }
}