Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, and Sunday are the names in English, followed by Monday again. A week like this is known as a planetary week. The Sun is the planet that most influences our daily lives with its movements around the Earth. It is responsible for nearly all of our seasons, and affects how much sunlight we receive. The Moon is the closest planet to Earth, but it doesn't influence day-to-day life very much. It is more of a cosmic companion than a solar one.
Friday is called Jiyu'e in Chinese culture because it marks the beginning of the weekend and summer vacations. In Japan, where Friday is also the first day of spring break, it's called "Umi no Hi." (the Sea's Day). This holiday was invented by the Japanese people who worked on ships sailing across the Pacific. They would spend Fridays away from port exploring new places and meeting new people, which is why Friday has become known as the "travel" day of the week in many countries throughout the world.
In Arabic culture, the name of the week is نهى الأسبوع meaning "the floating week". This comes from the idea that the moon causes ocean tides to rise and fall, just like it causes the tides on land.
The names of the days of the week in English appear to be a mishmash. Saturday, Sunday, and Monday are named after the celestial bodies Saturn, Sun, and Moon, respectively, while the other days are called after Germanic gods: Tiw's day, Woden's day, Thor's day, and Freya's day.
Saturday is the only regular daily appearance of any of these gods in modern literature. Shakespeare used them in several poems and plays, most often in reference to events on the Roman calendar. For example, when Rosalind asks why Ganymede was chosen for Jupiter's cup, she is told: "On account of his beauty, which would please even the king." Rosalind replies that such beauty is not fitting for a servant, but Jupiter agrees: "Tis not beauty that keeps the gods from rusticity, / But power, which power with beauty joins." (As you can see, the answer is not really important.) Jupiter goes on to say that he chose Tiw instead because Tiw owns the world on Mondays.
Sunday was originally called Sonday, meaning 'the seventh day'. It came to be regarded as the first day of the week because Christians believe that Jesus Christ resurrected on this day. However, this idea did not emerge until much later, probably around AD 300. Before then, people didn't think about doing anything special on Sundays, they just worked and played like any other day.
The week began to take form in Europe around 500 BC. It was probably originally based on religious festivals, but this meaning has been extended to cover all seven days of the week. The days were probably initially designated by number, with 1 being Monday and 7 being Saturday, but eventually became known by their current names.
In Germany, France, and Italy, Sundays were called "dies religiosus" or "jour des reliques" (religious day). This name came from a tradition that during this day saints' relics could be found in their churches wrapped in cloth stained with their blood. These relics were believed to have magical powers and to keep them safe, priests wore them around their necks at night.
Monday was called "dies monasterii" (monastery day) because it was traditionally when monks gathered together to hear sermons and to talk about events going on in the world. Monks wrote most of the books available in medieval times so this is where we get our idea of what a "weekend" is. They would also spend time praying together and discussing ways that Christianity could be improved.
The Romans called the days of the week after the Latin terms for the Sun, Moon, and the five known planets. Monday is luna septembria, "the seventh night" when the Moon is dark; Tuesday is luna octobris, "the eighth day" when the Moon is bright; and so on.
This system was probably introduced by Numa Pompilius, who was elected king in 715 B.C. According to some historians, he used to call the members of his family alphabetically, thus creating Monday through Sunday as we know them today.
However, it is more likely that this naming scheme was adopted by the Romans after they conquered a country if the language spoken there had no words or names for the days of the week. In that case, they would have to create their own names for them.
For example, they might name the days after the leaders of the conquering army or something similar. There's not much evidence for this theory but it's possible that it's what happened here. After all, the Romans did conquer Britain but they also absorbed many British traditions including the use of words or phrases in their daily speech.