PHP, Mysql, and Apache are probably some of the farthest softwares from being considered out of date, they are updated so frequently it's hard to stay up with the latest versions. If you think it's from the stone ages because it doesn't use a GUI ...well....
Apache is a webserver, this is what delivers the pages to the users visiting your website. A user types in the address of the server like "www.google.com" if you have purchased a name from dns like register.com and that send them to your sends them to your webserver in other words apache. Apache then see someone wants to view your site and sends them the html, php, and javascript code and whatever else your program might be developed in and thats pretty much all. This is why it's called a webserver because it is quite literally serving websites to people.
PHP is "PHP Hypertext Preprocessor" what happens is you write php code this is not html it's a whole different langugage. But what happens is the php program that gets installed on the server your on will run before your page is sent and parse all of your php before it is sent and all of it's functionality will be employed in the sent webpage. This is why you can not view source the php for a page because it has already been parsed before the page was sent over. So if you've worked with any programming language a good analogy for the php program your installing on the server machine would be the compiler and then the php that you write as code would be your language.
Mysql is a database management system and it's purpose is to organize and store information. Mysql is like a Microsoft Access database except that it doesn't have all the pretty graphics and is incredibly more robust. Once mysql is installed you will be able to create databases and use SQL to manipulate the tables of them and such. And then once you have php connected to it you will be able to access your database while creating pages allowing you pull out data such as addresses, names, messages, and posts which could all potentially be stored in the database.
To install all of these programs on a server you could use my favorite tutorial for doing so at http://www.ricocheting.com/server/apache.html
If you think this is to much and don't care a lot about the level of control then you could use something like XAMPP ( http://www.apachefriends.org/en/xampp.html )
And also if you are planning on creating a webpage that will be put up on the web then most webhosts have these programs already installed on them. Plus they have a lot of helper programs as well like phpmyadmin which will make mysql graphical.