Sometimes even the most seemingly simple questions have complicated answers. This is one of those. Fish are vertebrates. However, they are so diverse that they cannot be placed into a single smaller group that includes all of them. While all mammals are in a single class, fish are placed into five classes in two superclasses (the jawless fishes, including hagfishes and lampreys; and the jawed fishes, including cartilaginous, lobe-finned, and ray-finned fishes). Examples of four classes are shown below; there are no lobe-finned fishes in North America. So, in short, "fish" isn't really a category at all but a collection of smaller categories.