Standard Controller:

• Standard controller in apex, inherits all the standard object properties and standard button functionality directly.
• It contain the same functionality and logic that are used for standard Salesforce pages.
• It Can be used with standard objects and custom objects.
• It can be extended to implement custom functionality using extensions keyword.
• It provides a save method to allow you to persist changes.
• You’d use this when you have a singe object to manipulate.

Custom Controller:

• It is an Apex class that implements all of the logic for a page without leveraging a standard controller.
• Custom Controllers are associated with Visualforce pages through the controller attribute.
• Custom controller defines its own functionality.
• 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.
• You’d use this when your page isn’t dealing with a main object.