Learn Turkish: Bu kelime ne demek? - What does this word mean?

You are right! demek is the dictionary form of the verb "to say". However, in Turkish, asking what a word "says" is the standard way to ask what it means. The phrase ne demek literally translates to "what to say", but it is used as a set phrase meaning "what does it mean?".

Why isn't the verb conjugated here? It's just the dictionary form demek.

ne demek acts as a fixed phrase in this context. You don't need to conjugate it with a present tense suffix to ask for a definition. You can simply put the word you want to know about before ne demek to ask what it means.

Does Turkish not have a separate word for "meaning"?

There is a noun for meaning (anlam), and you could technically ask Bu kelimenin anlamı ne? (What is the meaning of this word?). But using ne demek is much more common and natural in everyday conversation when asking for quick definitions or translations.

I've heard people say ne demek when someone says "thank you". Is it the exact same phrase?

Yes! When someone thanks you, replying with ne demek! is like saying "Don't mention it!" or "It's nothing!" in English. It is a very common and polite response.