Quite simply, a deep link is a link pointing to a page on your website that isn't your home page. For example, if you have an e-commerce website selling tools, and you have a specific page listing hammers, a link from another website pointing to this specific page about hammers (e.g. yoursite.com/hammers.html) is a deep link.
Deep links give "link juice" DIRECTLY to the page they are linking to, so this is a good way to get a number of your web pages to rank well in the search engines
Deep links give your site a far more natural "link profile" to search engines. If you ONLY had links to your home page, it's not so natural because people like to link to interesting content - so they're more likely to link to a specific page than the home page of the website (where the users would then have to navigate to find the interesting content if it's not on the home page already). A hundred links to your home page and zero links to any other pages on your website isn't natural.