Фамилия means last name or surname. It does not mean family. Native Russian last names often end in -ов, -ёв, -ин or -ын for men; the feminine forms add -а. Thus when a husband and wife share a last name, the husband could say his last name is Смирнов and his wife will say her last name is Смирнова, and they are not lying, even though the names actually are a bit different. When Russians come to the US, the masculine form of the last name is usually made the official one, so the name Анна Смирнова might be transliterated as “Anna Smirnov.” That's not an error; it's just standard administrative practice.

To my observation, Russian women take their husbands last names rather less often than American women do, so it's not at all uncommon to find husbands and wives whose last names don't match. When children are born, however, they are always assigned a last name that matches the father's last name if the husband and wife are married.

Russian last names -ов, -ёв, -ин or -ын are not declined like standard nouns. They are declined like short-form adjectives. In the following table you will see a complete declension. Notice the endings with the light yellow background differ from standard first- and second-declension noun endings:

Here are some example sentences where you can see how the endings of the first name and last name sometimes match and sometimes don't. First some masculine examples:

And now some feminine examples:

And now some plural examples: