A Quality Backlink is a link that has clout and power. It's a link that has a pagerank which can flow to your site. They're links that are DO-FOLLOW and come from a page with limited outbound links.
Links that don't meet the criteria are;
long listings
FFA Lists
A 0 rank page
Alexa Rank in the millions
A site with no links from the outside to them.
Often you'll see some blog posting services where they'll post your link or to a number of blogs, or they'll post a blog comment for you on a number of sites, MOST times these sites are owned by the service you're purchasing from and they could be about anything. A blog that has details about car repair and then for some odd reason links off to "the best pet toy ever" is highly suspicious of being a PAID link, and wouldn't even get a second look by search engines.
Quality over quantity - just like food! You want to link from a reliable site, one who has followers or at least the same field that your busienss/site is in.