Search engines like Google, Yahoo and MSN are really just data bases. When you do a Google Search you are not searching the "world wide web" you are searching Google's data base. There are two ways to get in to these databases. One is to submit your site to the different search engines. In about 6 to 9 weeks the search engine will index your site. They have software that comes to your site and index every page and adds it to the database.
They come into your main web URL and follow the links to all the pages of your site. They capture key elements from the code on your page to the content. These are then stored in the data base. When someone puts a keyword in the search box the "algorithms" determine the best page or links to the best pages for your search.
The other way to get added to the search engines data base is to have the search engine software find you through a link to your site from another web site back to yours. The software, called "spiders" will periodically comeback and re crawl your site to see if you have updated it.
One important thing to know is that each page on your site is indexed individually and each page stands on its own. The ranking are based upon the combination of correct meta tags, relevant content to the keyword they are trying to get rankings for and link popularity. Mostly one way links back to their site from relevant sites.
As long as the search engine can index the site, clearly read the meta tags and content, the better. The big issues come when a site is built in flash with very little content (search engines cannot read or index "flash" sites. Also, if the bulk of the relevant content is in PDF format this is bad because the search engines cannot read PDF. If the search engines cannot index the relevant text there will be no rankings.
You all ready know this but "sites" are not ranked, individual pages are ranked for specific terms found in the content of that page as well as link popularity, (relevant links pointing back to that specific page) for the term that you are trying to get ranked on. This is why each page has to be giving very specific attention. Paid search
Paid search is when your ad shows up at the very top of a Google search or down the right hand side of the results page. These are called "sponsored ads". You pay for those positions. When every you click on one of those ads the owner of the ad pays the search engine. This is also called "pay-per-click". The amount you pay is determined by several factors including what you are willing to pay every time someone clicks on your ad.
Natural Search is when you type in a "keyword" and a link and a description shows up on left hand side of the search result page. The only way to get on the first page is to have very relevant content on your web site and links to your site from other relevant sites. This is a long process for people in it for the long haul. There are a lot of companies that claim to be able to get you on the first page of Google. That may be true if the "search term" is very specific and no one else would ever search for it but you or they are using a "black hat" method that could get you site banned from the search engine. Bottom line: Paid Search means you pay for your position. The benefit is, if you have the $$, you will get instant traffic. Stop paying and the traffic goes away. Natural search is free traffic but it is built over time. The advantage is, if done right, it can provide traffic for a long time.
Search traffic (paid or natural) is the BEST traffic to have because you are being found by folks who are specifically looking for what you have. It does not get any better than that.