There are two different Islamic days of observance that begin with "Eid." The first is Eid al-Fitr. The second is Eid al-Adha. These are two very different holy days for Muslims and neither bears any resemblance to the Christian holiday of Christmas. Christmas, of course, celebrates the birth of Jesus and is always, at least in the Western world, observed on December 25 (it is observed on January 6 in Asian and Near Eastern countries with Christian populations). While the date of the 25th of December remains constant, there is no consensus on exactly why that particular date is associated with the birth of Jesus. The fact that Christmas celebrates the birth of Jesus and is always observed on the exact same day of the year sets it apart from the Muslim holy periods of Eid al-Fitr and Eid al-Adha.
The fact that the Muslim holy month of Ramadan is ending with this writing could suggest that the student's question references Eid al-Fitr. This "Eid" is always held on the final day of Ramadan, which is a month of fasting among Muslims during the day, with daily break-fasts occurring at sundown each evening. Eid al-Fitr, as the final day of the holy month of Ramadan, is a day of celebration that marks the end of the month-long period of fasting. The significance of Ramadan--as no discussion of the importance of Eid al-Fitr can be complete without some reference to the importance of the month that precedes it--is that it commemorates the month in which Allah revealed the Islamic holy book, the Quran, to Muhammed.
Another distinction between Christmas and the holy month of Ramadan is in the calendars of each faith. As noted, Christmas is always observed on December 25 for reasons that remain historically vague. Ramadan is similarly ritually observed on a calendar and is always observed in the ninth month of the Islamic calendar, but its "official" commencement each year varies depending upon when key Islamic figures observe a new moon. As Muslims live in every region of the world, however, they may observe the start of Ramadan on different days because of the simple fact that the sky appears differently from various vantage points around the globe. Some Muslims follow the Saudi Arabian schedule, as it is within the borders of that kingdom where Islam's holiest sites are located. Others may observe the start of Ramadan based upon the presence of a new moon relative to their own geographic location.
Eid al-Adha is not associated with Ramadan. It is a different, and very important day of observance for Muslims. While Eid al-Fitr commemorates the end of the month-long period of fasting, Eid al-Adha commemorates something entirely different: Abraham's (Ibrahim to Muslims) determination to take the life of his beloved son Isaac as a demonstration of respect for and obedience to God. Because of its association with the Biblical story of Abraham's willingness to sacrifice his son, Eid al-Adha is called "the Feast (or Festival) of the Sacrifice." A whole lot of sheep lose their lives this time of year, as they are sacrificed.
In conclusion, there is no real similarity between the Christmas and either Eid. All are important to their respective followers. Beyond that, each is unique to its own monotheistic religion.