If you have a website, and even if you don't, you might be somewhat familiar with Apache HTTP Server. It is a program that runs as a service (aka daemon) on a computer. It has one, or sometimes many hundreds, of websites stored in its internal data folders. Its job is to receive incoming requests from the outside world and send the requested pages to the people or programs that asked for them. When you requested with your browser this page you are reading now, you requested it from the Apache web server that handles this website.
MySQL is very much like Apache Web Server, except that its purpose is to serve database data rather than web documents. It runs as a service on a computer. It has one, or sometimes many hundreds, of databases and their tables stored in its internal data folders. Its job is to receive incoming requests from the outside world, extract the requested data from its database tables, and send the data to the people or programs that asked for it.
Apache and MySQL are server programs that send information out to people (or programs) who request it, but if you want information, how do you send your request? You do it by using a separate "client" program. The client knows how to a) establish a communication channel with the server (which might be, and often is, on a separate and distant computer), b) properly format and send your request to the server, c) receive the data back from the server, d) organize and format it into a presentable form for you to view it.
That sounds very fancy, but confusion can easily be cleared up with just one example: What is the most popular type of client program in the world? A web browser! Now reread the previous paragraph, and you'll see that's exactly what it does. While Apache is a web (document) server, a browser is a web (document) client.
The same model holds true for MySQL. Since the purpose of MySQL Server is to send data out, you need a separate program that you use to connect to the server and get the data from it. Just as you can use a variety of different web browsers to connect to Apache Web Server, you can use a variety of different database client programs to connect to the MySQL server. Some of the client programs will be discussed later.
The key to the client/server model is that the two sides are handled by two separate and different programs, and I think this is the first and most important point of confusion that people might experience when trying to understand MySQL after working with another database program like Microsoft Access.