I have tried reading some articles, but I am not very clear on the concepts yet.
Would someone like to take a shot at explaining to me what these technologies are:
Tieme put a lot of effort into his excellent answer, but I think the core of the OPs question is how these technologies relate to PHP rather than how each technology works.
PHP is the most used language in web development besides the obvious client side html, css, and javascript. Yet PHP has 2 major issues when it comes to real time applications:
1) PHP started as a very basic CGI. PHP has progressed very far since it's early stage, but it happened in small steps. PHP already had many millions of users by the time it became the embed-able and flexible C library that it is today, most of whom were dependent on it's earlier model of execution, so it hasn't yet made a solid attempt to escape the cgi model internally. Even the commandline interface invokes the PHP library (libphp5.so on linux, php5ts.dll on windows, etc) as if it still a cgi processing a GET/POST request. It still executes code as if it just has to build a "page" and then end it's life cycle. As a result, it has very little support for multi-thread or event driven programming (within PHP userspace), making it currently unpractical for real time, multi-user applications.
Note that PHP does have extensions to provide event loops (such as libevent) and threads (such as pthreads) in PHP userspace, but very, very, few of the applications use these.
2) PHP still has significant issues with garbage collection. Although these issues have been consistently improving (likely it's greatest step to end the life cycle as described above), even the best attempts at creating long running PHP applications require being restarted on a regular basis. This also make it unpractical for real time applications.
PHP 7 will be a great step to fix these issues as well, and seems very promising as a platform for real-time applications.
You can easily use Node.JS in your web app only for real-time communication. Node.JS is really powerful when it's about WebSockets. Therefore "PHP Notifications via Node.js" would be a great concept.
See this example: Creating a Real-Time Chat App with PHP and Node.js
I have tried to make note about these and have collected and written examples from a java perspective.
HTTP for Java Developers
Reverse Ajax - Old style
Async Handling on server side
Reverse Ajax - New style
Server Sent Events
Putting it here for any java developer who is looking into the same subject.
In the examples below the client is the browser and the server is the webserver hosting the website.
Before you can understand these technologies, you have to understand classic HTTP web traffic first.
The server sends an event to the client when there's new information available.
The server and the client can now send each other messages when new data (on either side) is available.
Comet is a collection of techniques prior to HTML5 which use streaming and long-polling to achieve real time applications. Read more on wikipedia or this article.
Now, which one of them should I use for a realtime app (that I need to code). I have been hearing a lot about websockets (with socket.io [a node.js library]) but why not PHP ?
You can use PHP with WebSockets, check out Ratchet.