I am new to this SEO thing and I want to know in detail what Google page ranking is and what all this fuss all about.
I once came around the same topic, what I found is that Google started page ranking system pretty early (late 90s) when there were few others. Google unlike others, uses information stored in the back pages of a server to determine the rank of the site.
For search related terms, Google determines how many times a particular term appears in the page, and of course how many times a user has clicked a particular page for a given search term in a given area, that means different areas will give different search results.
And lastly, Google checks what links a page provides a checks their relativity for the determining of the ranks.
Bruce has posted a useful link about how PR works. In essence it is based on implicit "recommendations" of other web sites linking to yours. As a way of ranking sites it is flawed because placing a link on your web site to another is inviting visitors to leave the site and go somewhere else. Often that isn't a desirable thing for the owner of the originating site.
PR is just one of approx 200 signals Google use in deciding where your site will be listed in results for a specific search - and each of those signals may be based on 50 factors.
High PR is indicative that you got things right and there's likely to be a correlation between high PR and good search results but it is only an indicator. One ecommerce website I'm familiar with is the leader in its market in UK but only has a PR of 3 because there are few other high-rank sites that would have good reason to link to it (loads of low PR sites do) and because PR is based on links the PR is not a good indicator to use in this instance.
Note also that Google punish high PR sites found to be selling links (and there's a mechanism for reporting such sites).
From the perspective of a website owner one of your objectives is (probably) to be found in Google search results by people looking for your product/service/information. I guess that's what's behind your question.
The first question to ask is: What search phrases will be used by people looking for your product? In my opinion this is the biggest challenge for SEO. Logs and analytics will show you what phrases people have used to find your site but it won't show you what other people used and found, not your site but your competitor's.
There are tools to help with that. Finding those "right" phrases only puts you on a level playing field with everyone else trying to do the same. If you use those phrases in an appropriate manner you have at least eliminated a disadvantage but you want to be "better than..." not "the same as..." So it's time to look at the website from a user's perspective not Google's. Someone doing a search may look at 2 or 3 of the resulting links. You need to get them to stay on your page rather than one of those others. It needs immediate impact. It needs to scream "Yes this is what you need". It needs to download quickly.
SEO's love to "help you choose your search phrases" - their emphasis will be to find low usage but seemingly relevant phrases because that gives them the best chance to demonstrate "success". Look, they say, "we've got your phrase onto page 1" irrespective of the fact that nobody but you will ever search using that phrase.
If you want to optimise your website don't look for some kind of "magic bullet", there isn't one (although there are plenty of SEOs who will sell you one). Instead read Google's webmaster guidelines. Good quality, authoritative, informative, well designed, well structured content with no "tricks" intended to subvert Google's normal ranking processes should deliver results. Don't expect overnight success but organic growth. If the content is good, relevant high PR sites may link to yours and that endorsement is useful but it's still only 1 of 200 signals.
Measure SEO effectiveness not by search results position or number of hits but by positive actions taken by visitors - e.g. for an ecommerce site, the impact on sales (or, better, on profitability).
There are two pageranks; one is internal, one is public.
Public PR is a number from 0-10 that gives a rough guide as to how trusted (by google, at least), the domain is. It only updates once a year I believe, and basically means nothing. However, if you sell your site, it'll determine strongly how much you can sell for as it's beleived to correlate with googles internal ranking somewhat.
On the internal side, domains and URLs have a pagerank as a strong factor to where they will appear in the search results. A site with a high PR such as MakeUseOf.com will usually outrank someone random blog - though not neccessarily.
A page with a high PR will pass on more value to the URL it links to in turn: so if we at MakeUseOf link to your website, you'll see that page start to rank more than if someone random blog linked to your site.
Although its impossible to say what your internal PR is, in SEO circles we will always refer to some as helping you rank better; that just means trying to appear closer to position 1 in the results, and helping to build that internal PR value by various methods. Ultimately it comes down to writing good content, complying with google rules (not too many ads, no scammy links, no rewriting content), and producing content that people enjoy, read, and will interact with. Not complicated really.
This is probably one of the best explanations of what Page Ranking is all about.