OOP is the backbone of most programming languages so you will want to use it basically all the time when you have it fully understood.
Websites are a little trickier to give you good examples of so I'll try and give a generic outline and then a more web-based focus.
Classes should hold data that is related to each-over in a common sense name. For example a Car would obviously hold the Engine (another class), the colour, the name, the owner, etc. And within that the engine would hold it's own set of data that is relative to it (max speed, miles per gallon, etc.). This is the basis of holding your data together, next comes functions within classes.
Now while Car holds data relative to the car, Car also holds functions that the car would do, or functions that the car will use to interact with it's data. Getters(GetColour()) allow outside objects to get information about that specific car (as Car1 and Car2 could hold different colours). Setters are the opposite, to set object specific data. Then we have everything else. Car.start() which internally calls this.engine.start() which may set up some functions to start draining gas from the engine and allow the car to move.
Classes are all about having this easy to read, easy to understand, and easy to re-use. Once your Car class is complete you can re-use as many cars as you need and they are all nicely self-contained.
On to web
Firstly, I highly recommend you look into MVC architecture, which is where the bulk of web architecture is and makes OOP easier to understand for web. Basically your website is split into controllers (in charge of interracting with the DB and giving you back the page), models(Your database data, e.g a user) and your views (HTML with a little templating so your controllers can pass it dynamic data)
For example: You go to website/posts/Stanley , Your Router (another MVC thing) decides that this needs to be handled by the PostsController(a class). The posts controller knows that you want all posts by user Stanley, so it asks the Posts Model(a class)to retrieve all posts by Stanley, the controller then passes these to the view to fill out a template using your Stanley specific posts.
Going away from MVC a second,
With your example given, you will mostly use classes, I would assume, for data manipulation and DB access. For example if you stored blog posts in a database you may have a class Blog with data that matches your db rows (id, user_id, title, body) and functions to access that data to put in your HTML, but also to grab that from the database easily.
For example:
BlogPost::All() may return an array of every blog post newest to oldest.
BlogPost:By(user_id) may return an array of posts by a specific user.
BlogPost:Remove(id) may remove the blog post given it's id.
BlogPost:RatingAtLeast(starRating), get all blog posts with atleast a starRating rating.
You get the idea... This class would exist as you said above, to give you a client code way to access your database that is SPECIFIC to your use case and as BlogPost contains variables related to your BlogPost. So BlogPost::All() will return an array of BlogPost.