A search engine allows users to search for information on the internet by typing a phase (known in SEO as a keyword).
Examples of English language based search engines include Google, Yahoo and Bing, among others.
Whilst "Search Engine Optimisation" can sound a bit daunting, it basically means:
the art of altering the internal and external features of a website/web page, in order to increase its visibility in organic (i.e. un-paid) Search Engine results for phrases that are relevant to that particular website/web page.
SEO identifies keywords relevant to your website content, based on terms that people are looking for.
SEO optimises content to be well-written and contain relevant keywords so that search engines know what your page offers.
SEO implements a range of technical processes to enhance performance of a domain or page.
Facebook, Twitter, LinkedIn, Google+, Pinterest, etc.
Whilst social media can increase traffic to your website (through brand awareness, etc), it is not directly part of SEO. It is often considered and recommended as part of any good SEO strategy, however social media is mainly used as a marketing tool for businesses and is therefore comes under the umbrella of SEM (Search Engine Marketing).
A good SEO company will normally make recommendations on any obvious web page design issues (poor colouring, confusing menus, etc), however SEO mainly focuses on guiding relevant, good quality traffic to your web pages. Keeping users on your website once SEO has got them there is normally in the realm of a Marketing department, however a good SEO company can be of great use in determining which pages users are leaving from, and can set up tests to explain why.