Let's consider English and Hindi - geographically (almost) the most distant members of this family:
Monday: The day of the moon. In Hindi, this is "Somvaar", with "Som" meaning the moon, and "vaar"being day.
Tuesday: This one is not so obvious - named after Tyr, the Norse god of war, who in the Roman panatheon, is Mars(pictured). In Sanskrit (the predecessor to modern Hindi), Mars is "Mangal", hence "Mangalvaar".
Wednesday: The day of the norse god Woden, better known as the Roman messenger god, Mercury. The Hindi name - "Buddhvaar". ("Buddh" should not be confused with the Gautama Buddha.)
Thursday: Thor's day. Based on Jupiter, king of the Roman gods, and the largest of the planets. In Hindi, this planet is commonly called "Guru" - meaning "large". The name "Brihaspati" is also used - hence "Guruvaar", and the less common "Brihaspativaar".
Friday: For Frige, the Norse goddess of love. Venus to the Romans, and "Shukra" to India. "Shukravaar".
Saturday: No points for guessing this one - Saturn's day. The Hindi equivalent here is "Shanivaar".
Sunday: In Hindi - "Ravivaar". "Ravi" is a another name for the sun.
What's even more interesting, is that not only are the days named after the same planets, but the gods for each planet serve similar roles in both mythologies.
This commonality exists across almost all Indo-European languages. Though, some languages have tweaked a few days to more logical names- e.g. Wednesday becomes Mittwoch("Mid-week") in German, Friday is sexta-feira("sixth-day") in Portuguese.