With an adult cat you'll have a very good idea of the cat's temperament straightaway. A kitten may behave one way in its youth change significantly (and not always for the better!) as it grows up. An adult cat's personality will be more consistent, and you'll be able to gauge any traits and quirks through its interaction with the staff of the shelter.
Easy does it
Another reason to pick an adult cat over a kitten, frankly, is it's less work. Kittens need lots of attention to keep them stimulated and out of trouble. An adult cat is likely to be much easier on an owner who doesn't have the time or energy to entertain a kitten.
Adult cats found in shelters are seldom there because there is something wrong with them or with their personality. Most of the time, the owner's living conditions have changed or allergies have arisen in the original home. Usually these are friendly, socialised cats who would love to be part of a new family.
The first thing to look for when adopting a cat is a personality that is a good fit for you. The shelter staff will be able to give you a good idea of the cat's temperament. They'll be able to tell you if she is good with other animals or better by herself. They'll also tell you if she's very clingy or more laid back. If you're bringing a new cat home to a house that already has cats and dogs in it, look for one that has lived with other animals before. Regardless of her background, any cat will be happier in a good home than in a shelter.
Get to know each other
Be sure to spend some time alone with the cat or cats you want to bring home. Ask the shelter staff if there is somewhere quiet, away from the cattery, where you can be with the cat. Being in the shelter may have been extremely stressful and frightening for the cat so it may take a few minutes before she calms down and allows her true personality to shine through.
Most shelters will already have done a thorough health check and the cat is likely to be up to date on all its vaccines and already spayed. Still, give the cat a good once over from nose to tail to try to spot any health concerns.
Check the small print
Some shelters require that you sign a contract when you're ready to adopt. Be sure to read this thoroughly. It may have requirements such as not giving away the cat as a gift. If you're planning on giving the cat as a gift, arrange for the prospective owner to visit the shelter and participate in the adoption.
In general pick the cat (or cats) to which you feel the strongest attraction. Good luck and congratulations on the new addition to your family.