From my understanding the second Eid that you are speaking of is actually not related to Ramadan~ I found the following at Wikipedia:

Eid ul-Fitr or Id-Ul-Fitr (Arabic: عيد الفطر ‘Īdu l-Fiṭr), often abbreviated as simply Eid, is a Muslim holiday that marks the end of Ramadan, the month of fasting. Fiṭr means "to break the fast" and therefore symbolizes the breaking of the fasting period. On the day of the celebration, a typical Muslim family is awake very early and then after praying the first normal everyday prayer, is required to eat in a small quantity, symbolizing the end of Ramadan. They then attend special congregational prayers held only for this occasion in mosques, in large open areas, stadiums or arenas. The prayer is generally short, and is followed by a sermon (khuṭba). Worshippers greet and embrace each other in a spirit of peace and love after the congregational prayer. After the special prayers, festivities and merriment are commonly observed with visits to the homes of relatives and friends to thank Allah (swt) for all blessings.

For Muslims, Eid ul-Fitr is a joyous occasion with important religious significance, celebrating of the achievement of enhanced piety. It is a day of forgiveness, moral victory, peace of congregation, fellowship, brotherhood and unity. Muslims celebrate not only the end of fasting, but also thank Allah (swt) for the help and strength that they believe he gave them throughout the previous month to help them practice self-control. It is a time of giving and sharing, and many Muslims dress in holiday attire.

Eid al-Adha (Arabic: عيد الأضحى ‘Īd al-’Aḍḥā) is a religious festival celebrated by Muslims worldwide as a commemoration of Ibrahim's (Abraham's) willingness to sacrifice his son Ismael for Allah. It is one of two Eid festivals that Muslims celebrate. Like Eid ul-Fitr, Eid ul-Adha also begins with a short prayer followed by a sermon (khuṭba).

Eid al-Adha is 4 days long and starts on the 10th day of the month of Dhul Hijja (ذو الحجة) of the lunar Islamic calendar. This is the day after the pilgrims in Hajj, the annual pilgrimage to Mecca in Saudi Arabia by Muslims worldwide, descend from Mount Arafat. It happens to be approximately 70 days after the end of the month of Ramadan.

Eid-ul-Adha (Adha Eid) has other popular names across the Muslim world, such as Eid el-Kbir in Morocco, Algeria, Tunisia, Egypt and Libya.

While Eid ul-Adha is always on the same day of the Islamic calendar, the date on the Gregorian calendar varies from year to year since the Islamic calendar is a lunar calendar and the Gregorian calendar is a solar calendar. The Lunar calendar is approximately 10 days shorter than the Solar calendar.[1] Each year, Eid ul-Adha (like other Islamic holidays) falls on one of two different Gregorian dates in different parts of the world, due to the fact that the boundary of crescent visibility is different from the International date line.