It's been said a million times in a million threads but I can say it again...
Class design in this game is so good that every class is "wanted" in group and raid situations for that unique something that they bring to the table.
You may not be first choice in every situation, but then, no one is!
The most important keys to being sought after as a character (in no specific order) are these:
~ Play your class well - learn what makes you special and work that angle till you have it down pat! If you're playing a dps character, find the tricks that will let you do the most dps (for your class) and focus on making that happen consistently. If you're a tank, learn your taunts and learn everything you can about how to absorb damage. And if you're a healer, learn what makes your healing class and style different and how you are the best at healing. (Different healers work best with different tanks)
~ Keep your gear and your spells up to the best quality they can be. If your spells are all at app1 & app2, no one is gonna want to carry you but if your "important" spells are kept up at ad3 or ma1 and the rest are at least ad1, people will take you more seriously. The same goes for armor and weapons - when you change tier, re-gear! If you use poison or ranged ammo, always carry the best quality you can afford for your current level of skill.
~ Build a network of friends in game or even better - join a good guild. This will give you a pool of players to draw from when you are looking for a group or when you need to fill a slot an an existing group.
Several of the people I play with all the time, maintain huge friends lists and when we are forming up a group to do an instance run they will "hit the bricks" looking for someone to fill the slot. It's always nice to get a tell saying "we're gonna go do [blank] and we're looking for a [blank], wanna come along?"
If you learn your class and keep your gear up, those tells will often find you!