SEO ("organic search" or "natural search") is a web marketing technique aimed at improving the performance of web content in search engines. Performance is typically measured by testing the appearance of a website for targeted search terms, and more specifically, ranked positions in search for important keywords.
Broadly speaking, SEO techniques focus on:
content optimisation, ensuring that desirable content is provided and that it is properly attuned to current trends and behaviours in the target demographic;
link building, a process of encouraging (or deliberately placing) links to the promoted content from external websites and, more recently, social media.
Each of the above is generally considered as a separate specialism within the broader discipline of SEO, though any SEO campaign will usually involve all three.
Black Hat / White Hat
Historically, SEO has been divided into two categories: black hat and white hat.
Though the dichotomy is necessarily somewhat crude, "black hat" refers to techniques that intentionally exploit (or "game") shortcomings in the design of search engine algorithms, or techniques that otherwise violate search engines' terms of service, for the sole purpose of manipulating search rankings.
"White hat" refers to techniques which aim to optimise website design and content within terms of services, typically building long-term value and carried out as part of a wider marketing strategy.