Search Engine Optimization (SEO) is a set of methods aimed at improving visibility and ranking of a website in search engine listings. This means that internet users find this site through unpaid search engine listings, rather than pay per click advertisements (which - by the way - can cost up to several tens of dollars per visitor!). If SEO is executed correctly, the website will appear more frequently and more prominently in search results on Google, Yahoo, MSN, AOL and other search engines.
This means more actual prospective customers, ready to buy your specific product, rather than merely passers-by, who will not convert to actual sales or increase in your business' visibility.

Increasing your ranking is not an overnight process. Here is how SEO professionals (including us) work through it:

Enter a general description of your business' industry into google.com. If it's real estate, you'll learn that there are approximately 334,000,000 web pages containing this phrase and an average internet user doesn't usually go beyond page three of search results (that's 30 out of those 334 millions). Do you want to get your own website on one of those first three pages? Then you better start optimizing your website both on- and off-the-page!

"On the page" basically means web design, which is in 100% control of a website designer and thus easily modifiable (assuming that person has Search Engine Optimization knowledge). It includes optimizing titles, contents (text), tags, graphics, etc.
Please note that on-the-page optimization is easier to perform but has lower effect on rankings and positions.

"Off-the-page" optimization mostly deals with popularity and importantance of your website on the internet. There are many ways of improving those and making search engines display it higher than the competition in search results. Some of those methods involve: adding listings, conducting link exchanges,etc.