Linkshells are essentially chat channels, often with a specific purpose.

They're less of a commitment then a Free Company, but it's a way to keep in contact with like-minded people.

Popular kinds of Linkshells include Hunt Linkshells, focused on relaying information about locations of Hunt Marks in the open world
FC alliances, intended to facilitate cooperation between FC's and for filling in parts for any dungeneering/raids that happen
Static Coil LSes, as a dedicated chat channel for static groups whenever you're not in a party
And just general chat channels.

You can be in multiple linkshells at a same time. Just remember which one is which so you don't sound completely silly when talking into the wrong one!