Glyfair said:
There are two primary things that affect ranking. The primary one is the number of links to the site it finds with those words (lots of "4th edition" links to the site). The other one is by paying them money for a better ranking.
There have been known instances of deliberate manipulation of this (such as a certain political figure's website showing up when a search for "stupid idiot" was entered).
Google never sells better rankings on its search results, though some other search engines do.
For those who are interested or care, I do know the basics of Google's crawler algorithm, as we spent a lecture examining it and its brethren in a course I took.
Other factors include the prominence of the words and their font and size (they are considered more important when set aside in large font at the top than when found randomly in the body of the text), the authority/hub dichotomy common to Kleinberg-esque algorithms (though Google's PageRank isn't exactly like Kleinberg's).
But the basics you list are correct--high profile links from authority sites on the topic or popular hub sites will indeed increase a page's ranking--
The simplified version: Picture a random guy on the internet who clicks links to surf through pages randomly based on prominence (though at superhuman speed), with some probability 'd' that he will get bored and start over at a random page on the internet. That's Google's web crawler. The sites that wind up getting him what he wants get higher rankings.
In the words of the algorithm's creators "PageRank can be thought of as a model of used behaviour. We assume there is a “random surfer” who is given a web page at
random and keeps clicking on links never hitting “back” but eventually get bored and starts on another random page. The probability that the random surfer visists a page is its PageRank. And, the d damping factor is the probability at each page the “random surfer” will get bored and request another random page."
Congrats! 
