Search Engine Optimisation has two distinct area's the first being On-page optimisation and the second, off-page optimisation. On-page optimisation is what you can actually do to your website that will affect your ranking on the search engines. This includes changing your title tags, H1 Tags etc. Search Engine Optimisation (SEO) like anything else adheres to the 80/20 rule, whereby on-page optimisation accounts for 20% of search engine rankings.
The other 80% comes from link building, which is by far the hardest part when it comes to SEO. Link building is getting other sites to link back to your own website. Like everything else in the world, links have varying degrees of quality, you get really poor quality links which can actually harm your website or you can get excellent links which will help your rankings tremendously. Obviously, the best quality links are the hardest to obtain. For your information, the best kind of links to get, are links that come from Universities or Government websites. Search Engines love these links and if you do manage to get one your website will almost certainly help get your website on to the first page within a couple of months.
I will list the ways in which a person can obtain links for their website.
1) Buy Links - The better the link, the more expensive they are.
2) Ask For Links - Either via Phone or Email.
3) Link Exchanges - Swap Links with other webmasters.
4) Register with Directories - Register your site with a number of directories.
5) Write Articles - Submit your articles to article directories like Ezine.
6) Write Press Releases - Submit to sites like PRweb.
7) Use Viral marketing techniques - Use sites like pay per post (link baiting).
I would suggest doing all of the above when you first start trying to do SEO on your own site. A good mix of links is always good, try not to buy too many "cheap & cheerful" links as these will not do your site much good.