Lamp or “Deep” is the symbol of knowledge. Lighting the lamp of knowledge within us means to understand and reflect upon the significant purpose of each of the five days of festivities and to bring those thoughts in to our day to day lives.

The first day of Diwali:
The first day of Diwali is called Dhanvantari Triodasi or Dhanwantari Triodasi also called Dhan Theras .It is in fact the thirteenth lunar day of Krishna Paksh (the dark forthnight) of the month of Kartik. On this day, Lord Dhanwantari came out of the ocean with Ayurvedic medicine (medicine which promotes healthy long life) for mankind. 
This day marks the beginning of Diwali celebrations. On this day at sunset,Hindus should bathe and offer a lighted deeya with Prasad (sweets offered at worship time) to Yama Raj (the Lord of Death) and pray for protection from untimely death.
This offering should be made near a Tulsie tree (the Holy Basil) or any other sacred tree that one might have in their yard. If there is no sacred tree, a clean place in the front yard will suffice.

The second day of Diwali:
The second day of Diwali is called Narak Chaturdasi. It is the fourteenth lunar day (thithi) of the dark forthnight of the month of Kartik and the eve of Diwali. On this day Lord Krishna destroyed the demon Narakasur and made the world free from fear. On this day, we should massage our bodies with oil to relieve it of tiredness, bathe and rest so that we can celebrate Diwali with vigour and devotion.
On this night, Yama Deeya should NOT be lit. The Shastras (Laws of Dharma) declares that Yama Deeya should be offered on Triodasi night with Prasad.
The misconception that Yama Deeya should be offered on the night before Diwali came about some years ago when the fourteenth lunar day (Chaturdasi) was of a very short duration and caused Triodasi to extend into the night before Diwali. Some people mistook it to mean that because Yama Deeya was lit on that night, that it should always be lit on the night before Diwali.
This is absolutely not true. It is advisable that one consults with a learned Pandit or Hindu Astrologer for proper guidance on this matter.

The third day of Diwali: 

This is the day when worship unto Mother Lakshmi is performed. Hindus cleanse themselves and join with their families and their Pandit (priest) and they worship the divine Goddess Lakshmi to achieve the blessings of wealth and prosperity, the triumph of good over evil and light over darkness.

The fourth day of Diwali.

On this day, Goverdhan Pooja is performed. Many thousands of years ago, Lord Krishna caused the people of Vraja to perform Goverdhan Pooja. From then on, every year Hindus worship Goverdhan to honour that first Pooja done by the people of Vraja.
It is written in the Ramayan  that when the bridge was being built by the Vanar army, Hanuman (a divine loyal servant of Lord Rama possessing enormous strength) was bringing a mountain as material to help with the construction of the bridge. The call was given that enough materials was already obtained. Hanuman placed the mountain down before He could have reached the construction site. Due to lack of time, He could not have returned the mountain to its original place.
The deity presiding over this mountain spoke to Hanuman asking of His reason for leaving the mountain there. Hanuman replied that the mountain should remain there until the age of Dwapar when Lord Rama incarnates as Lord Krishna in the form of man. He, Lord Krishna will shower His grace on the mountain and will instruct that the mountain be worshiped not only in that age but but in ages to come. This deity whom Hanuman spoke to was none other than Goverdhan (an incarnation of Lord Krishna),who manifested Himself in the form of the mountain.
To fulfill this decree, Goverdhan Pooja was performed and is continued to be performed today.

The fifth day of Diwali.

The fifth day of the diwali is called Bhratri Dooj. This is the day after Goverdhan Pooja is performed and normally two days after Diwali day.
It is a day dedicated to sisters. We have heard about Raksha Bandhan (brothers day). Well this is sisters day.
Many moons ago,in the Vedic era, Yama (Yamraj, the Lord of death) visited His sister Yamuna on this day. He gave his sister a Vardhan (a boon) that whosoever visits her on this day shall be liberated from all sins. They will achieve Moksha or final emancipation.
From then on, brothers visit their sisters on this day to enquire of their welfare.
This day marks the end of the five days of Diwali celebrations.

This is also known as Bhai fota among Bengalis. Bhai fota  is an event especially among Bengalis when the sister prays for her brother's safety, success and well being.