Dog groups are human categorizations of dog breeds according to specific criteria. Dogs belonging to any one group are supposed to have something in common, although there is a lot of diversity within each dog group.
The categorizations of dog breeds into groups are a very old idea and the criterion that is used is old and doesn't measure up to the needs we have today. To make it even more confusing, the groups vary a bit from country to country. Most common is that these groups are divided by the dog type like working dog, toy dog and so on. But the working dog can also be a companion dog and vice versa, so the differentiation can be blurred sometimes.
One word of advice: It is generally a better idea to pick a dog based on the breed rather than the group to which it belongs. It is with the breed that you get accurate information about personality, skills, health issues, grooming requirements, etc. All of these things vary within the major groupings. Still, it is important to understand the basic features of each dog group so that you can narrow down your interests.