When you first get a dog, you might want to consider basic behavioral training. This teaches dogs to behave well around both people and other animals. It may include some basic commands, but the goal is to make them good citizens as dogs. Some of this training may deal with behavioral problems like excessive barking, chewing, or housebreaking. This is a very basic type of training that is appropriate for younger dogs that are just getting started.
Obedience training is focused on making dogs obedient to their owners through the use of commands like sit, stay, lie down, etc. It is slightly more advanced than the behavioral training, but it will usually include some training to resolve any behavioral problems before they start. This type of training is also appropriate for younger dogs as well.
Agility training is for dogs that will participate in dog sports, like obstacle courses, racing or jumping. This is definitely a more advanced type of training which already assumes that the dog understands basic commands. The handler is not allowed to touch or reward the dog during the competition, so there will have to be a strong connection between the owner and the dog through voice and physical gestures. Although it is true that any dog can learn these skills, certain breeds are more suited to these tasks than others.
Dogs are capable of learning a wide variety of skills just like people. There are dogs that learn how to herd, hunt, do search and rescue work, assist the disabled, or even work with law enforcement. These skills are actually like vocational training for dogs because it will mean that they will have a career helping people in some way. In these programs, dogs learn very specific techniques to hone their senses and communicate with people, but the programs are also rigorous and time-consuming.
Dogs are truly amazing creatures as they can easily go beyond basic obedience training and actually save lives. All dogs can learn regardless of their background, but certain skills are going to be easier for specific breeds due to their physical traits. The sky is the limit as long as you can find a qualified instructor to train your dog.