How do Jews celebrate each of the 8 days of hanukkah?

With the exception of the exact number of candles lit, each of the eight days is celebrated the same.

During Hanukkah, Jews light the Chanukiah, also called a Hanukkah-menorah (which is an 8-branched candle holder) each night, singing, and eating traditional foods such as latkes or jelly donuts. Parties are also common, as well as chocolate coins, called "Hanukkah gelt".
One candle is lit on the first night, with an additional candle added until all 8 candles are lit on the eighth night. During the candlelighting, 2 blessings are sung (3 on the first night), and often other songs are song after they are lit.
The only traditional gift in Hanukkah is the gelt given to children.
In more detail:
At nightfall one candle is added each night, starting from the right and going to the left. Then the candles are lit from the newest one on the left and then one by one to the right. This is done every day; thus on the 8th day 8 candles are lit. People light their menorah every night at or after sundown.

Special Hanukkah additions are added to the prayers: Al-hanisim (regarding the miracles) in grace after meals and in the thrice daily prayers

Hallel - the Thanksgiving psalms during morning prayers

Special Torah reading

A popular Hanukkah game is "dreidel" - a four sided top

Dairy products are popular Latkes (potato pancakes) Sufganiyot (jelly donuts)

Giving out Hanukkah money (gelt) to children is a popular custom