Any hyperlink that is linking to a web page on your website from external web page is called a backlink.

In English, any link from another website with a different domain (hisdomain.com) to a web page on your site (yourdomain.com) is called a backlink. The backlink could be from entierly different domain, or it could be a web page in your sub-domain.

How do you check your backlinks?

There are many paid and free tools in the market that you can use to find out who is linking to you. I will just suggest a few FREE ones I most oftenly use for your reference.

Google Webmaster tools

Google Webmaster Tools shows backlinks on your sites that Google can find on their indexed pages, but for some reason it did not display the backlinks when captured the screen ... server error?

SERP (Search Engine Result Page) refers to the page that display the result of the search query. More related, more useful and more trustworthy indexed web pages will appear higher in the SERP, thus  the term SERP Ranking.

PageRank is a propriety algorithm developed by Google to determine the usefulness of an individual web page. Ranking scores from 0-9, where 0 being the lowest and 9 being the highest rank.

Characteristics / Implications

PageRank and relevance of these links are to the topical theme the targeted page is about. The algorithm of PageRank is constantly changing.

Background Information

Before Google was created, Yahoo and other search engines index web pages manually -- i.e. if you have a website and if you wish your website to be listed in Yahoo, you will have to submit the web address (URL) to Yahoo, and Yahoo will send someone to take a look at your web page and manually abstract crucial information from your site and list them on their search engine database. In the later years Yahoo and many other major search engines implemented a paid advertisement scheme that lists sponsored listings higher in the search results.