Learn Icelandic: Hetjan fór til jötunsins þegar bardaginn hafði byrjað. - The hero went to the giant when the battle had started.

Just like in English, it combines the past tense of the helper verb hafa (to have) with the supine form of the main verb. Here, we use hafði (had) and byrjað (started). It is used to make it clear that the battle started before the other past action of the hero leaving.

Why doesn't the verb form byrjað change to match the masculine noun bardaginn?

Because it is acting as the supine (sagnbót) in a perfect tense, not as an adjective describing the noun. Whenever a verb directly follows a form of hafa to create the perfect or past perfect tense, it takes this fixed form ending in -ð or -t and never changes, regardless of the subject's gender or number.

Why is the word for "the giant" written as jötunsins instead of just jötunninn?

The preposition til always forces the following noun into the genitive case. The base word is jötunn (giant). In the genitive case, it becomes jötuns. When you add the definite article suffix to mean "the" giant, it becomes jötunsins.

Could I use hvenær instead of þegar for the word "when" in this sentence?