Learn Italian: Questo regalo significa molto.

Questo means "this" and must match the noun it goes with. Because regalo (gift) is masculine and singular, we use the masculine singular form questo. If you were talking about a feminine noun like a house (casa), you would say questa casa.

What is the dictionary form of the verb significa?

It comes from the regular -are verb significare (to mean or to signify). Because "this gift" is a third-person subject ("it"), we drop the -are and add the "he/she/it" ending -a to get significa.

Since the lesson is about question words, how can I ask "What does this mean?"

You can use the question word cosa (what) combined with this verb! Just say Cosa significa questo? (What does this mean?).

Will molto ever change to molti or molta in sentences like this?

No. Here, molto is an adverb meaning "a lot" because it describes the verb significa and tells us how much it means. Adverbs never change their endings. If you used it to describe a noun (meaning "much" or "many"), it would change to match that noun, like in molti regali (many gifts).