Learn Russian: Какая ваша фамилия?

Что is used when asking 'What is that object?' (like Что это?). But when you ask for specific information that describes a noun—like a name, color, or type—you use a form of какой (which/what kind). Because you are asking 'which surname is yours', you use the feminine form какая.

Where is the word for 'is' in this sentence?

In Russian, the verb 'to be' (am, is, are) is usually left out in the present tense. You just put the words next to each other. Literally, this sentence is just 'What your surname?'.

Does фамилия mean 'family'?

No, it is a 'false friend'! Фамилия specifically means 'surname' or 'last name'. If you want to say 'family', the Russian word is семья.

Why do both какая and ваша end in -а?

In Russian, describing words must match the gender of the noun they connect to. Фамилия is a feminine noun (because it ends in -я), so it requires the feminine forms какая (what/which) and ваша (your).

Could I say твоя instead of ваша?