It is unknown how long days have been grouped into weeks of seven days, but some of the earliest records begin with the Hebrews. In Hebrew, the days are simply numbered one through seven, with seven being the Sabbath (Shabbat). Greek, Russian and Portuguese are similar, having religious names for Saturday and Sunday (though Russian has the Sabbath as day six, not seven).

Greek:
Deftera (second)
Tritê (third)
Tetartê (fourth)
Pemptê (fifth)
Paraskeuê (preparation)
Savvato (Sabbath)
Kyriakê (Lord's Day)

Portuguese:
Segunda-feira (second)
Terça-feira (third)
Quarta-feira (fourth)
Quinta-feira (fifth)
Sexta-feira (sixth)
Sábado (Sabbath)
Domingo (of God)

Russian:
Ponedelnik (after "do-nothing")
Vtornik (second)
Sreda (middle)
Chetverg (fourth)
Pyatnitsa (fifth)
Subbota (Sabbath)
Voskresenye (Resurrection)

Latin was a bit more creative, naming the days after the visible heavenly bodies, also the source for the names of the Roman gods. The European languages that don't follow the Hebrew numbering scheme usually use a variation of the Latin names.

Latin:
Lunae dies (Moon)
Martis dies (Mars)
Mercurii dies (Mercury)
Jovis dies (Jupiter)
Veneris dies (Venus)
Saturni dies (Saturn)
Solis dies (Sun)

French is closely related to Latin and, like the other Romance languages, retains similar names. The dies ("day") part has been contracted into a -di suffix and the last two days are derived from alternate Christian Latin names for Sabbath and God's day.

French:
Lundi
Mardi
Mercredi
Jeudi
Vendredi
Samedi (from Latin Sambati dies (Sabbath))
Dimanche (from Latin Dominius (of God))

English borrows from Latin and French, but it is primarily a Germanic language. Unlike French, it retains the Latin source for the "religious" days of Saturday and Sunday. But it localizes some of the Roman god names into their Anglo-Saxon/Nordic equivalents: Mars, Mercury, Jupiter, and Venus are replaced with Tiw (Tyr), Woden (Odin), Thor, and Friga (Freya).

English:
Monday
Tuesday (Tiw (aka Tyr))
Wednesday (Woden (aka Odin))
Thursday (Thor)
Friday (Friga (aka Freya))
Saturday
Sunday

German, like English, is Germanic with a heavy Latin influence. Dienstag, from "assembly day", replaces Mars. Mittwoch, or midweek, replaces Wednesday. Donnerstag means thunder day, a reference to Thor, god of thunder. The references to Roman or Norse gods were probably deliberately replaced as the Germans were Christianized.

German:
Montag
Dienstag (assembly day)
Mittwoch (Midweek, from mittauuechun, coined by Notkur, a monk, circa 1020 AD, to replace pagan reference to mercury)
Donnerstag (thunder day, reference to Thor, god of thunder)
Freitag (Freya, as in English)
Samstag/Sonnabend (Samstag is probably from French Samedi, Sonnabend is "Eve of Sunday")
Sonntag