Unlike a fellowship, which is a short term grouping of players, a kinship is a long-term organization of players with an internal structure of a leader and officers. Kinships are sometimes called guilds in other games. If you open your social panel, you'll see a tab for kinship. If you join a kinship, that tab will show all the members of the kinship who are online, and you'll also get a kinship chat channel, so you can talk to members of your kinship easily. In that sense, a kinship is like an extended friends list with a means of talking to one another. The real advantages of kinships, though, is that (depending on the characteristics of the specific kinship) they usually offer to members in the kin and are a solid group of friends to run instances and raids with. That help may be in the form of help on quests and instances, advice and guidance on how to play your class well, crafted items for kinmembers, and so forth.
Different kins have different focuses. Some are very focused on leveling together and helping newer players learn their classes, while others are more end-game focused and spend most of their time raiding and running instances. Your server kinship forum will have posts by kins that are recruiting new players, which will tell you something about the nature of the kinship, it's goals and rules. Some kins are very family friendly and have younger players, so they specify no bad language. Others are more...colorful ;p Many kins use external voice programs like ventrillo or mumble to talk to one when they are not grouped up in game, so it's a fun way to get to know people.
As far as how to join one, I would suggest looking through the kinship forum for your server to see who is recruiting and also group in game with different people and make note of what kinships the ones you like are in. Often if you like one person from a kin, you'll enjoy spending time with other members too. An officer from the kin will send you an invitation to join the kin. Accept the invite and you will be in the kin. If it turns out not to be to your liking, you can leave kin from the kinship tab in the social panel (it's generally considered polite to tell an officer why you are leaving), and try to find one that fits your own needs and interests.
A kinship is a player-formed grouping of people. Unlike a fellowship, a Kinship can have hundreds (or thousands!) of other members. Once you join, you are a member of the kinship until you either quit or are expelled by an officer. Other games call these "guilds", I think.
When you join a kinship, you get access to a dedicated chat channel for just you and the other members. Depending on what the people in the kinship choose, you may:
Ask for advice or help. Most folks are MUCH more likely to help kin members than random strangers
Ask crafters in the kinship to craft things for you, sometimes for free
Be able to travel to the Kinship House, and perhaps access items stored there
Get access to a voice server that your kinship runs outside of the game
Find kinship-only auctions on the Auction house
Communicate with your kinship outside of the game via a web site or forum
Kinships do typically expect members to contribute their time and resources as well. For example, if you are asking for help with quests, you should also offer your help when others ask. If you are asking for crafted items, you should contribute materials, or offer to craft items
Many kinships also organize events. They can be anything; from standing in front of the Prancing Pony playing music to taking on Saruman in the Tower of Orthanc raid. Players who like to do group content (like raids) tend to join kinships because people who play together regularly get more coordinated over time and can more easily master difficult content.
Whether a kinship is what you want is up to you. If you are looking for one, reading the forum here for your server can tell you what kinships are currently recruiting.