Let's make an analogy to a car. A car has several components that make it useful. You have an engine to provide power, but just an engine will not make it go. We need a way to transfer that power to the wheels, controlling its speed. We also need to control the directon. So we have transmission, which controls if the engine is connected to the wheels, and at what ratio. It even controls if the wheels will go backwards in relation to the engine rotation.
Then we have the steering system. This allows the wheels to turn in different directions to make the car go in different directions. There may be a device to assist the user in turning the steering wheel to turn the driving wheels.
Now we may want to stop. For this we have a brake. This puts resistance to the wheels making it harder for them to turn which makes the car slow down.
In the same way, we have various components to a computer system, both software and hardware.
SQL is indeed a standardized way to make requests from a database, but it has been so widely used, that it often describes the database itself. MySQL is very popular, as it is opensource. OpenSource means that one can get to freely and without cost, look at the design of the database and make changes to it that might improve it.
PHP is a website scripting language. A programming language who's purpose is to provide web pages. PHPBB is a bulletin board system written in PHP.
Now such a bulletin board needs a way to store and retrieve information, mainly the posts that we make. This is stored in a database. The bulletin board retrieves that information using SQL calls to the database.
Linux is an operating system that is also opensource. It could be equated (in functionality) to Windows. Windows is not opensource. You'd have to pay Microsoft to the nose in order to glimse at their source code.
Apache is an open source web server. Its job is to provide web pages that you and I request. Apache usually runs on Linux, but has been ported to other operating systems.
So, PHPBB runs on Apache, which runs on Linux. PHPBB uses MySQL (in this case, there are others) to retrieve data that we want to see. MySQL also runs on Linux. It is possible to have MySQL and Apache to be running on two different machines, just as you could have a Records department and a Sales deparment either in the same building, where a sales person would walk to Records to get information he needs, or in separate buildings, where a telephone call might be made to request and retrieve the information.