A backlink is any link pointing at your site from somewhere else.
The more relevant a site, with high page rank will carry the most weight.
Page rank is Google's method for determining page authority and rank. From what I remember, it is a logarithmic scale of (I think) 8 times. That means a link from a page rank 1 site is worth eight times more than a zero. A PR 2 one is worth 64 etc.
Relevance is very important too. For carpet cleaners, a link from inside a paragraph on the home page of say, the NCCA or TACCA would be really powerful as would any link from a high PR carpet cleaning related site.
An in context link like above is far better than a low quality one from a "links" page.
Social sites also give backlinks (for instance the Google + homepage is a PR9). As these are supposedly harder to fake, they are (allegedly) more and more important. Lot's of activity on social sites should indicate that the site is popular.
The anchor text (the blue, underlined bit) also says a lot about your site and used to be vital for ranking sites. What the Penguin updates did was look for unnatural looking (fake) link structures.
A "natural" pattern would have lots of "click here" or "visit site" or
www.carpet cleaner.co.uk type etc
If a site had a hundred backlinks and ninety nine had "carpet cleaning your town" as the anchor text, it was pretty certain that something naughty was going on. These were the one's that got slapped.