A team of people you play with regularly. A team you compete along side with.
Could be a group of friends you enjoy playing with. Could be a group of people that are good at something in particular.
I'm sure someone could explain it better. But in essence it's just a team. Like a sports team for video games, I guess. Some clans are very sports team-esq, in where they actually have specific roles to play, and coaches. While most clans are just groups of people that play together often.
A clan can be various different things.
It may be a handful of people who know each other in real life who just like to play together and have a common tag name.
It may be a group of gamers who enjoy similar gaming styles or game modes and have met through the game and have found they work together, so they stick together.
It may be a group of people who just want a large pool of people to game with, and enjoy the concept of bringing people together and organizing them under one flag that they started and watching it grow.
It may be a larger more organized clan who want to game with not only similar minded people but also hold certain requisites to attain membership such as kill/death ratio, communication/teamwork ability, etc. Or they may not even care about personality so long as the membership has a certain skill level or better.
Any way you put it, when you decide to join a clan make sure you look into them a little.
The way I run my squad is I play with groups of my people, and let our (usually ) wins speak for themselves, show maturity and humor in lobbies. Usually that sparks people's interest when they see the common gamertags, and they may inquire. If I play at least a handful of games with someone and I like playing with them, and against them, I might invite them to party up with us. If it goes well, I'll begin telling them about our clan and see if it sparks an interest.
Whether you're looking at joining a clan or looking for members to add to a clan, something important to keep in mind is that you're choosing the people you will be playing with every day (assuming that clan unity is strong). I look at it like adding a member to my gaming family. So you want to choose carefully and not be grouped with people who will make you look bad or give you a bad reputation once you strap on that clan name.