I am trying to create a web app that will access to my database, and be able to SELECT, INSERT, UPDATE and DELETE records. I am new to using databases and PHP, and have looked at other questions trying to make a breakthrough in my understanding, so please be kind and don't mark this as a duplicate.
The general answer to your question is MVC framework. If you want to build web app, it will make your life easier in many ways: routing, communication with database, managing data, splitting frontend from backend. There are a lot of them out on the internet, the one I can recommend is Laravel. Although you can use any with which documentation will you be satisfied. Symfony, Zend, CodeIgniter and much more.
From your question it looks like you lack information about PHP and general app building mechanisms, so trying to make your own base for app from scratch may result in disaster and discouraging you to continue. If you have some free time, the easiest way to gain knowledge should be books. They will guide you from the very first letter in code till the end. Just be careful to buy a good book, so you won't spend time learning bad habits (just read book's comments).
Although, I will try to briefly answer the questions you stated. It shouldn't be hard to find out more on the internet.
The most common way to communicate from frontend (html) to backend (php) is by forms. You create a form with action targeted on PHP file script (it may be even the same page, but in general we avoid that). You can use for example POST method. Thanks to it, after user fills it and clicks submit button, PHP script will receive all data via POST. In script you can use $_POST, which is a global collection of data sent with a form. Let's look at the example:
<form action="this.php" method="POST"> <input type="text" name="name" placeholder="Please enter your name"> <input type="email" name="email" placeholder="Please enter your email"> <input type="submit"> </form>
in this.php file there will be available: $_POST['name'] and $_POST['email']. You can now work on them and finally put them into database.
There is also GET methods. It allows you to send data via url parameters. Let's say you have too much data in database, and you need to split it into pages. You can create a href "mytestpage.php?page=2" that will represent what page you are currently on. You can read this attribute with $_GET['page'].
If you use plain PHP, always be sure to take precautions. Have in mind that user can pass dangerous data and violate your scripts.
Read more: http://api.jquery.com/jquery.ajax/