What search engines are we talking about?
The Big Three are the ones that you might use yourself most of the time: Google, Yahoo and Bing.
With almost half a million visitors per month, Yahoo.Com is the most visited of the three. However, Yahoo is a portal that is used for many, very many reasons, from Yahoo Email or Messenger, to News updates, hotel and car rental systems, the social network Yahoo Answers, and much more.
Google at the other hand is a search engine and a search engine only. All of its visitors use Google for the search functions only.
In search, Bing has now overtaken Yahoo. It has become the clear Nr 2 as a search engine... but wait... again there are changes on the way.
But do not forget there are still very many other search engines, such as Ask, Lycos, Altavista, Netscape and Cuil. Basically every site or portal with a search function can be called a search engine.
Basic Vocabulary
If you turn to Google or Yahoo and perform a search, then this search engine will return a page with results. This is a Search Engine Result Page or SERP.
In SEO the aim is to rank high in these SERPs. You perform such a search by making use of a search term.
The search engine will then start looking for this term in as many web pages as possible... It will look for Key Words.
It can be just one or two words, or even a long combination of words, in which case we speak about Key Phrases.
The SERPS of for example Google are divided into two parts: You have the free listings on the left, and the paid advertising on the right (and top left).
The goal of SEO is to rank high in the free listings - to avoid the sometimes expensive Paid Advertising, also named Pay-per-click or key word advertising.