1

一月一日から一月七日、または一月十五日まで、新しい年のお祝いをします。

I would understand this sentence if the part in bold was omitted: New Year is celebrated from January 1 to January 15. What is "一月七日、または” for?

2

または means or.

So, the sentence means New Year is celebrated from January 1 to January 7 or January 15.

一月一日から一月七日、または一月十五日まで、新しい年のお祝いをします。

Other examples:

アメリカ、または日本に行きます。

I'll go to the US, or Japan.

このテストはスミス先生、またはジョンソン先生に渡してください。

Please hand this test to Smith-sensei, or Johnson-sensei.

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