"pinging" is basically an automatic notification to a search engine (not Google.com!) that a page has been updated and it should be recrawled. Specifically, this is relevant for social properties like blogs, forums and other web 2.0 sites that are indexed by several search engines that keep a record of the page and the date.
The big search engines (i.e., Google) index those sites quite frequently so your chances of increasing your site rate of index is higher. Google also has its own blog index that you can ping.
Pinging backlinks, usually means that you ping a page that is pingable (like a blog post comment) which links to your site (backlink).
When you ping backlinks, it increases your chances of getting that backlink indexed and get its link juice (if applicable).
There are a lot of different pinging services out there, and in a lot of the platforms they're already set to ping new posts to various directories but here's two popular ones: