Learn Hardcore Icelandic: Konan kaupir flík.

The -n is the definite-article suffix, so konan means the woman. In Icelandic, definiteness is shown by adding a suffix to the noun:
• Feminine singular: -in (kona → konan)
• Masculine singular: -inn
• Neuter singular: -ið
If you drop the suffix (kona), it becomes indefinite: a woman.

How would you say a garment in Icelandic?

You have two options:

Omit the article entirely and simply say flík (this is common).
Use the indefinite article ein, which must agree in gender, number, and case. In nominative singular feminine you’d say ein flík, and in accusative singular (as object) you’d add case ending: eina flík.

What case is flík in, and why does it look the same as the dictionary form?

Here flík is the direct object of kaupir, so it’s in the accusative singular. Many feminine nouns ending in -a have identical forms in nominative and accusative singular, so you don’t see any change.

Why is the verb kaupir, and what is its infinitive form?