SEO is short for "Search Engine Optimization." It's a collection of tactics designed to help your web pages rank better for the keyword phrases you're targeting.
The main difference between SEO and say, Pay-Per-Click advertising, is that the visitors you receive from your listings will not cost you anything. These free listings are also called "organic listings" and there are usually ten such listings per page.
Your listings are triggered whenever someone searches for a particular term (aka keyword phrase) using a search engine. The search engine does its best to return the most relevant web pages for the search performed.
Your goal, then, is to help the search engines understand what your web page content is most relevant for. The clearer you communicate your relevance, the more likely it is that you'll receive good search engine rankings.
Generally speaking, there are two main SEO techniques to perform to each page of your site. They are:
1. On Page SEO
On page optimization includes anything that is done to your actual web page. As each page is under your control, you can ensure that it is optimized for the keyword phrases you believe are most relevant to your content.
2. Off Page SEO
Off page SEO is more challenging as you can only control so much. Any webmaster can link where they please and if they've linked to your site using an undesirable bit of text, you may suffer the consequences.
However, you can control the majority of the off-page work to be done and it's important you do. Linkbuilding is generally considered to be the most important "off page" factor and ignoring it will hamper your ability to rank well for the terms you are targeting.