A local yarn store, commonly abbreviated "lys" in the knitting and crochet communities, is a yarn shop with a physical location. It is the opposite of an online-only yarn store, which can often afford to charge less for its wares as it has fewer overhead costs. (Having said this, many local yarn stores sell products online in addition to in their physical locales, and many small yarn and fibre brands sell their products online out of small studios that don't operate as commercial stores in the traditional sense. The online shops I'm referring to above are larger web-only retailers that sell big-name brands for discount prices.)
In the age of the internet, arguments in favour of patronizing local yarn stores include fostering one's local crafting community and supporting local business. (This is true of all local businesses, but supporting a local yarn store vs. purchasing yarn more cheaply online is a bit of a rallying cry within the craft community. Besides that, people actually refer to their "local yarn store" using that particular phrase in everyday conversation.)
Many local yarn stores offer classes for all levels of crafters and covering basic concepts and specific techniques. A local yarn store's owner and staff are also typically passionate about their work and are able to answer questions and make recommendations, not only about their yarn products but about patterns and techniques. Most local yarn stores worth their salt will have staff members who will be willing to help you out if you show up with a project you've messed up or a pattern you can't figure out.
Crafters refer to their local yarn stores in a different manner than people might refer to their local grocery store or movie theatre. "Your" local yarn store might not be the shop closest to you but is rather one that you treat as your crafty home base because you prefer its selection, get along with its staff or just feel like it's your yarn store. (A good analogy would be your favourite comic book shop or used bookstore.)
The abbreviation "lys" is commonly used online, on crafting blogs and on communities such as Ravelry, though people tend to just say "local yarn store" (or shop) when they're speaking aloud.
(As Urban Dictionary notes, "lys" can also mean "learn your shit," a patronizing admonition to someone who doesn't know what they're doing.)