Age. Two common questions people have is "Can I start training my puppy?" and "Can my older dog do agility?". The answer is yes to both questions, within reason. To compete in agility your dog needs to be at least a year old, but you can start to train your puppy earlier than that. Keep in mind, however, that your dog's joints are still forming, so jump heights should be about elbow height for the first year. Puppies will also be more awkward and their attention span shorter. So be very patient and keep the lessons brief. The same principles apply to older dogs. Their joints are weaker, and they will tire more easily, so lessons should be shorter. The jump heights might have to be lowered. Many of the organizations now have special classes for older dogs. AKC calls these "preferred" classes, which is a class that feature lowered jump heights. But be careful that you do not get so caught up into competing that you are not sensitive to your dogs aging needs. Dogs are so anxious to please that they may push themselves one day, and pay for it the next.
Breed. You will see all shapes and sizes of dogs (and people) doing agility.... this is one of the wonderful things about the sport! You may think to yourself, "my dog can never do that, he's too small", but in agility classes and trials, your tiny little guy will have lowered jumps, a lower table, and will be competing against other dogs his size. For example, he might be running in the 6" or 8" jump height division, and that class will have lower time limits (because it takes longer for small dogs to run a course than big dogs). Large breeds also can do agility, but their bulkier size requires negotiating some of the obstacles more carefully. Also be aware that some breeds take more patience than others. Even a bull dog can do those obstacles, providing you find the special treat or toy that really motivates him. Some dogs are highly motivated simply with praise.
Obedience. Can crazy out-of-control dogs do agility? Yes, as long as you make your training sessions more exciting than the other things around your dog that typically makes him go crazy! See the article on "Problem Dogs" for tips on how to do this. It is not necessary that your dog has formal obedience training if you want to do agility just for fun. If you want to be more successful at it, however, your dog must know some basic obedience commands such as stay, sit, and lie down. Heeling is helpful. While taking a class is invaluable, you can (at home) train obedience and agility at the same time. Some people train their dogs to sit for the first time using the pause table obstacle, for example.