What is the difference between a standard controller and a custom controller in Salesforce Apex?

Difference between Standard Controller & (Custom) Controller
Visualforce controllers are used to access a visual force markup associated with particular link or button. These controllers can be used to provide access to the data that should be displayed in a page, and can modify component behavior. There are number of standard controllers exists for each Salesforce object which provides the functionality similar to custom controller.

A custom controller  is the user defined an Apex class that implements all of the logic for a page without leveraging a standard controller. Whenever the visualforce pages need to run entirely in system mode and it does not enforce the permissions and field-level security of the current user the custom controller works. You can have controller extension to each controller in visualforce.

A controller extension is moreover an Apex class that extends the functionality of a standard or custom controller. The controller extensions can be used in following scenarios:

You want to leverage the built-in functionality of a standard controller but want to override one or more of them for custom specification such as edit, view, save, or delete.
You can add new actions.
Visualforce page which respects user permissions.
The controller extension class executes in system mode, but if it extends a standard controller, then the logic from the standard controller does not execute in system mode. Instead, it executes in user mode, applying the sharing rules, field level security, permissions of current user.