Some examples:
With each example, any item or person in the subset is also found in the original set (but not vice versa). For example, any dog is an animal but not the other way around.
If you were to draw a venn diagram, then the larger circle is the set of all animals, while the subset of all dogs is a smaller circle drawn completely inside the larger one.
Those examples listed above is of course not the complete list of set & subset combos. Feel free to get creative and think of others.