Social Integration is clearly a loose term that defines a whole spectrum of social site development techniques. Many are designed to provide viral content to the masses while others are simply to allow your site the ability to interact with social websites such as Facebook and Twitter.
Auto posting is when you set up a number of continuances that allow your new posts and articles from your website to automatically be posted on various social websites as they are being published. For example, when you publish a new article on your website, instead of going to each social website and posting a link to that article, it will be posted automatically to the social sites of your choosing without requiring you to do any of that extra legwork through the use of plugins, extensions or cron jobs.
These days it's common for people to include a Facebook Comment system on their websites. This is usually due to the fact that Facebook has its own way of managing the spammers. However, one of the most common reasons is the social interaction and Search Engine Optimization that these comment systems provide because people are not only commenting on posts on your website, but they are also spreading the word about your website to people on their social networks.
Adding "like me on Facebook" buttons and "follow me on Twitter" buttons (among others) is a great way to let more people know that you exist. It allows people to share your content with other people on their networks and tells the search engine that your content is what people are looking for. This enables the search engines to know that you are providing relevant content for people on a regular basis and boosts your search engine ranking in the process.
One of the most annoying things that I can think of is having to register on a website. maybe I'm just lazy but I think that a lot of other people feel the same way. It really get's monotonous after a while when you find yourself doing the same old' routine of entering a username, a password, a password again, and then a bunch of other information that you feel is completely unnecessary just to enter various sections of a website. However, by implementing a social login system, people can simply register and login to your website at the click of a button by using their Google, Facebook, Twitter, or LinkedIn accounts (to name just a few of the more popular ones).
There are many other types of social integrations such as message boxes, social sharing links, Etc. But this gives you the general idea. Sharing is not simply a term used to impress the kids anymore. It's a way of making your content go viral and your web presence become a popular conversation if you put enough into it. It's all about social networking and the idea is to network it around you, your products and your services.