Learn Danish: Hvalpen er to år gammel. - The puppy is two years old.

In Danish, you don't use a separate word for the. Instead, you attach the article to the end of the noun. The word for a puppy is en hvalp. By moving en to the end of the word, you get hvalpen (the puppy).

Do I have to use the word gammel when talking about age?

No, it is optional. Just like in English, you can say Hvalpen er to år gammel (The puppy is two years old) or simply Hvalpen er to år (The puppy is two).

Since the number is two, why doesn't the word år have a plural ending?

It is actually in the plural! In Danish, short neuter words (words that take et instead of en) usually don't change their spelling in the plural. Because it is et år (a year), the plural remains exactly the same: to år (two years).

Why is gammel in its singular form when it comes right after to år (two years)?

The adjective gammel describes the subject of the sentence, which is hvalpen (the puppy). Since there is only one puppy, the adjective stays singular. If you were talking about multiple puppies, it would become plural: Hvalpene er to år gamle.