What algorithms i can use from machine learning or Artificial intelligence which i can show via web site
I assume you are mostly concerned with a general approach to implementing AI in a web context, and not in the details of the AI algorithms themselves. Any computable algorithm can be implemented in any turing complete language (i.e.all modern programming languages). There's no special limitations for what you can do on the web, it's just a matter of representation, and keeping track of session-specific data, and shared data. Also, there is no need to shy away from "calculation" and "graph based" algorithms; most AI-algorithms will be either one or the other  (or indeed both) - and that's part of the fun.
For example, as an overall approach for a neural net, you could:

Implement a standard neural network using python classes
Possibly train the set with historical data
Load the state of the net on each request (i.e. from a pickle)
Feed a part of the request string (i.e. a product-ID) to the net, and output the result (i.e. a weighted set of other products, like "users who clicked this, also clicked this")
Also, store the relevant part of the request (i.e. the product-ID) in a session variable (i.e. "previousProduct"). When a new request (i.e. for another product) comes in from the same user, strengthen/create the connection between the first product and the next.
Save the state of the net between each request (i.e. back to pickle)

That's just one, very general example. But keep in mind - there is nothing special about web-programming in this context, except keeping track of session-specific data, and shared data.
I am die hard fan of Artificial intelligence and machine learning. I don't know much about them but i am ready to learn. I am currently a web programmer in PHP , and  I am learning python/django for a website.
Now as this AI field is very wide and there are countless algorithms I don't know where to start.
But eventually my main target is to use whichever algorithms; like Genetic Algorithms , Neural networks , optimization which can be programmed in web application to show some stuff.
For Example : Recommendation of items in amazon.com
Now what I want is that in my personal site I have the demo of each algorithm where if I click run and I can show someone what this algorithm can do.
So can anyone please guide which algorithms should I study for web based applications.
I see lot of example in sci-kit python library but they are very calculation and graph based.
I don't think I can use them from web point of view.
Any ideas how should I go?