What is Visualforce in Salesforce?

Visualforce and Lightning both offer ways to design and create custom interfaces for Salesforce.
Visualforce was designed to follow a Page-Centric model. This means that the intent of Visualforce was to create something that was your full page interface with Salesforce. When the user needed to perform some kind of operation, like Save a record, it would send that request to the Salesforce servers and then reload the entire page with the new state of the UI. All backend processing is done with Apex Code on the Server-Side. You have the option to inject JavaScript into the mix to handle some of the Client-Side processing, but it isn't the default interaction methodology. Visualforce was also primarily built for Desktop. When Visualforce was first launched back in 2008 the iPhone had been out for about a year. The concept of designing things "mobile first" hadn't yet been realized, this would come a few years later.
Visualforce also uses an HTML-like markup language for designing the pages and Apex code to handle the database operations. Here is a breakdown of what you have with Visualforce pages:
Visualforce Page - HTML-like markup language used to design the layout of the page.
Standard Or Custom Controller - Apex Code to handle Server-Side operations executed from the Visualforce page (EX: CRUD operations).
Optional: CSS Styling - Style your Visualforce page to look a specific way within the header or by file reference.
Optional: JavaScript - Used to handle Client-Side processing or to be coupled with CSS for a cosmetic revamp of the UI.
Optional: Apex Extensions - These are used to perform logical operations that are not housed within the standard or custom controller.