Setting the facts straight

In my amateurish attempts to state certain biological facts, I have used the terms like "family", "species", "cousins" incorrectly. I have looked up some of these terms and I found that the terms are more specific than I was aware of. I like to point out that occasionally scientist do make mistakes so certain animals or group are wrongly classified but they do try to set the record straight when new information is found.Below are very simplified explanations and they are not able to handle exceptions or controversies.

The smallest group of a living thing is called a species. All members of a single species can mate naturally and produce fertile offsprings. This is a reasonable explanation but it is not a perfect one. Their offsprings would also have the ability to mate with members of that species. 
There are several species that appear to have a very recent common ancestor. This group is called Genus. The members of a Genus must also have a distinction between its members and other living things.

There will be different Genus which have certain common characteristics. This different Genus can be group together to form a family. Members of a family also need to have a common ancestor. That is why the legless lizard is not a snake even though we assume all legless reptiles to be snakes.