Learn Russian: Какая это собака?

The question word has to match the gender of the noun it describes. Since собака (dog) is a feminine noun, you must use the feminine form какая. If you were asking about a masculine noun, like a cat (кот), you would use the masculine form какой.

Since собака is feminine, shouldn't это be эта?

Not in this case! Here, это is acting as a standalone word meaning "this is" (or in a question, "is this"). When used to introduce or point out a noun, это stays in this fixed "neuter" form regardless of the noun's gender. You would only use the feminine эта if it were directly attached to the noun to mean "this dog" (for example: Эта собака большая - This dog is big).

Does какая mean "what" or "which"?

It can mean both "what kind of" and "which" depending on the context. In this sentence, you are most likely asking about the breed or characteristics of the dog ("What kind of dog is this?"), but you could also be asking someone to identify a specific dog from a group ("Which dog is it?").

Where is the verb "is" in this sentence?

Russian usually drops the present tense verb "to be" (am, is, are). The sentence literally translates to "What kind this dog?" but in English, we naturally add the word "is" to make it "What kind of dog is this?".