Sunday:

Hebrew: Yom Rishon, First Day -- counting to Shabbat: Sunset Saturday to nightfall Sunday

Middle English sone(n)day or sun(nen)day
Old English sunnandæg "day of the sun"
Germanic sunnon-dagaz "day of the sun"
Latin dies solis "day of the sun"
Ancient Greek hemera heli(o)u, "day of the sun"

Monday:

Hebrew: Yom Sheini, Second Day -- counting to Shabbat: Sunset Sunday to nightfall Monday

Middle English monday or mone(n)day
Old English mon(an)dæg "day of the moon"
Latin dies lunae "day of the moon"
Ancient Greek hemera selenes "day of the moon"

Teusday:

Hebrew: Yom Shlishi, Third Day -- counting to Shabbat: Sunset Monday to nightfall Teusday

Middle English tiwesday or tewesday
Old English tiwesdæg "Tiw's (Tiu's) day"
Latin dies Martis "day of Mars"
Ancient Greek hemera Areos "day of Ares"
Tiu (Twia) is the English/Germanic god of war and the sky. He is identified with the Norse god Tyr.
Mars is the Roman god of war.
Ares is the Greek god of war.

Wednesday:

Hebrew: Yom R'vi'i, Fourth Day -- counting to Shabbat: Sunset Teusday to nightfall Wednesday

Middle English wodnesday, wednesday, or wednesdai
Old English wodnesdæg "Woden's day"
Latin dies Mercurii "day of Mercury"
Ancient Greek hemera Hermu "day of Hermes"
Woden is the chief Anglo-Saxon/Teutonic god. Woden is the leader of the Wild Hunt. Woden is from wod "violently insane" + -en "headship". He is identified with the Norse Odin.
Mercury is the Roman god of commerce, travel, theivery, eloquence and science. He is the messenger of the other gods.
Hermes is the Greek god of commerce, invention, cunning, and theft. He is the messenger and herald of the other gods. He serves as patron of travelers and rogues, and as the conductor of the dead to Hades.

Thursday:

Hebrew: Yom Chamishi, Fifth Day -- counting to Shabbat: Sunset Wednesday to nightfall Thursday

Middle English thur(e)sday
Old English thursdæg
Old Norse thorsdagr "Thor's day"
Old English thunresdæg "thunder's day"
Latin dies Jovis "day of Jupiter"
Ancient Greek hemera Dios "day of Zeus".
Thor is the Norse god of thunder. He is represented as riding a chariot drawn by goats and wielding the hammer Miölnir. He is the defender of the Aesir, destined to kill and be killed by the Midgard Serpent.
Jupiter (Jove) is the supreme Roman god and patron of the Roman state. He is noted for creating thunder and lightning.
Zeus is Greek god of the heavens and the supreme Greek god.

Friday:

Hebrew: Yom Shishi, Sixth Day -- counting to Shabbat: Sunset Thursday to nightfall Friday

Middle English fridai
Old English frigedæg "Freya's day" composed of Frige (genetive singular of Freo) + dæg "day" (most likely) or composed of Frig "Frigg" + dæg "day" (least likely)
Germanic frije-dagaz "Freya's (or Frigg's) day"
Latin dies Veneris "Venus's day"
Ancient Greek hemera Aphrodites "day of Aphrodite"
Freo is identical with freo, meaning free. It is from the Germanic frijaz meaning "beloved, belonging to the loved ones, not in bondage, free".
Freya (Fria) is the Teutonic goddess of love, beauty, and fecundity (prolific procreation). She is identified with the Norse god Freya. She is leader of the Valkyries and one of the Vanir. She is confused in Germany with Frigg.
Frigg (Frigga) is the Teutonic goddess of clouds, the sky, and conjugal (married) love. She is identified with Frigg, the Norse goddess of love and the heavens and the wife of Odin. She is one of the Aesir. She is confused in Germany with Freya.
Venus is the Roman goddess of love and beauty.
Aphrodite (Cytherea) is the Greek goddess of love and beauty.

Saturday:

Hebrew: Yom Shabbat: Sunset Friday to nightfall Saturday: 18 minutes before sunset Friday to nightfall Saturday: Havdalah

Middle English saterday
Old English sæter(nes)dæg "Saturn's day"
Latin dies Saturni "day of Saturn"
Ancient Greek hemera Khronu "day of Cronus"
Saturn is the Roman and Italic god of agriculture and the consort of Ops. He is believed to have ruled the earth during an age of happiness and virtue.
Cronus (Kronos, Cronos) is the Greek god (Titan) who ruled the universe until dethroned by his son Zeus.

Yom Tov:

In the times of the Holy Temple, the high court (Sanhedrin) would declare each new month (Rosh Chodesh) only after witnesses would come and say that they saw the new moon. This is a special power that God gave the Jewish people to control time, and is irrespective of any scientific knowledge. When the Sanhedrin declared the day to be Rosh Chodesh, they would send messengers to notify everyone which day is the first day of the month, so that they could correctly calculate the Yom Tov, or other occasions.

Those who lived far from Jerusalem, where the messengers could not reach before the 15th of Nissan (Pesach) or the 15th of Tishrei (Sukkot), would keep two days Yom Tov, out of doubt regarding which day the witnesses saw the new moon.

Nowadays, we do not have the Sanhedrin declaring Rosh Chodesh, and everything follows a preset calendar established by Rabbi Hillel II in the 5th century CE. Yet we still keep two days Yom Tov (Sukkot, Pesach and Shavuot) in the Diaspora, in commemoration of those days when we had two days Yom Tov in the Diaspora. (Maimonides - Kiddush HaChodesh 5:5)

On another level, some want to suggest that a second day Yom Tov was added in order to make a distinction for those living in the Diaspora: that for a Jew, not residing in the Holy Land is in a sense an aberration.

The reason that Yom Kippur is only one day even in the Diaspora is that it would be too difficult for people to fast for two days straight.

On Rosh Hashana (which is always the first day of Tishrei - i.e. Rosh Chodesh), we sometimes had two days of Rosh Hashana in Jerusalem as well. If the witnesses did not arrive the first day, they would keep Yom Tov that day, just in case they'd show up in middle of the day; and the next day. Therefore, by rabbinical decree, even in Jerusalem, we observe nowadays two days Yom Tov. (Maimonides - Kiddush HaChodesh 3:9, 5:7-8)

The Three Pilgrimage Festivals: Passover, Sukkot and Shavuot.

It was only when the Temple was standing that people were required to appear three times annually and bring an offering (Shulchan Aruch O.C. 117:1; Nodeh BiYehuda O.C. 94; Chasam Sofer Y.D. 233; Yabia Omer O.C. 5:15; Tzitz Eliezer 10:1)

Nevertheless, citing the Midrash from Shir HaShirim, some opinions maintain that coming today to see the Temple Mount and the Western Wall still applies on the pilgrimage festivals (Ran â€" Ta’anit 7a; She'alat Ya'avetz 1:87; Yechaveh Daas 1:25; Teshuvah Kol Mevaser 2:10).

Whichever view is adopted, today tens of thousands of people again make the pilgrimage to the site of the Temple and the Western Wall during these festival days.

Like the days of the week, other than Shabbat, The Hebrew months don't have names in the Tanach. They have numbers, counting from the month of Nissan, which is described as "the first month" at Exodus 12:2.

I Kings 6:2 refers to the month of Iyar as the "month of Ziv." The word "ziv" is an adjective and means "radiance." Rabbi Eliyahu Kitov explains that it is called "radiance" because in this month the sun is in full radiance. Similarly, the Jewish people came into full "radiance" in this month, for they were made ready to receive the Torah during this month.

I Kings 6:38 refers to the month of Cheshvan as "the month Bul," related to the word "baleh" which means, "withers," and the word "bolelin" which means "mixed." It is described in this fashion since the grass withers in this month, and the grain is mixed for the household livestock. The Radak explains that the word "bul" is related to "yevul" which means produce, since plowing and planting begins in this month.

Other names we use today are Babylonian in origin. They were adapted by the Jews some time during the Babylonian Exile, circa 400 CE. Ironically, the month of Tammuz is the name of an idol which appeared (via optical illusion) as if it was crying. This was achieved by putting soft lead into its eyes, and by kindling a small fire inside, which would melt the lead. This explains the reference in Ezekiel 8:14: "There were women sitting, causing the Tammuz to cry."

There are other opinions about the name of this month. Rashi says that the name Tammuz is an Aramaic word meaning "heat," since it is a hot summer month.

Another interesting note: Tammuz-17 was the name of the Iraqi nuclear reactor destroyed by Israel in 1981. It was so named because the 17th of Tammuz is the day that Jerusalem was sieged prior to the destruction of the Temple by Nebuchadnezzar, and Saddam Hussein was known to fancy himself as the heir to Nebuchadnezzar's fallen dynasty.

Even though the names of the months are linguistically speaking Babylonian, they were adopted by the Jews with the understanding that they were Divinely inspired names. They are laden with kabbalistic nuances. Based on this, the Sages expounded the names of the months - e.g. Elul is an acronym for "ani ledodi vedodi li" (I am to my beloved, and my beloved is to me”), and Nisan is the month of "nissim" (miracles).

Sources:

Crowl.org
JewFaq.org
AISH.com

We Are:
AllFaith.com
LearnEmunah.com
BeitEmunah.com