What happens on this day? Is there a ceremony or celebration among Muslims?
Best i can share with you from my experience.
Eid is celebrated after the month of Ramadan (name of a lunar month) being the month the Quran was revealed.
During the month Muslims fast (obligation), abstain from food water and sex from day break to sunset. They also increase prayer (non obligation but of good things to do) mainly with community or personal prayer at night time (terabi).
The desire, effort and discipline helps bring some change in their lives and Eid is a celebration of G-d goodness to man, or if you like the appreciation of what He gives is magnified.
Eid begins with sunset as lunar days begin then (Jews & Muslims), in the following morning after sunrise Muslims gather and pray Eid prayers, then eat plenty. Many invite people to their houses for a feast and try to get around accepting the invite of others.
Some celebrate Eid over three days.
There are other aspects not covered in this post, i guess one of the more significant to Christians would be the sharing of lamb, goat or cow (maybe some other acceptable clean animal). Some give the whole animal to the poor, some divide it in thirds, one to their own family one to friends and one third to the poor. (the poor are seen as beggars and those who do not ask). It is said the piety of one doing doing this what goes to Allah. In this light many people today of the Islamic religion donate money over seas and tinned meat is given to people.
Hope this helps answer your question, please remember this just what i understand and should not be taken as absolute fact for people of the Islamic religion are varied and the accepted scholars of the Islamic religion may answer better.
Pray well and stay well