We have two holidays in Islam (not counting Ramadan which is a time of worship for us), Eid al adha and Eid al fitr. These are times when Muslims contact family members and greet each other, a time for rest, and a time for celebration. There is also worship on these days (Eid prayer in the morning followed by a Khutbah (speech)).
There are some Sunnah's to do during Eid (Sunnah's are things the Prophet did for Allah's reward that we can do) such as eating an odd numbered amount of dates in the morning, recting "Allahu ackbar Allahu Ackbar Allahu Ackbar, La illaha illah lah, Allahu ackbar Allahu ackbar, wa lillahil Hamd," if I recall till Eid prayer, taking different paths onto the mosque and on the way home (because the angels pray for those who do that along the paths and it will result in more blessing and more reward), etc.
So these are things Muslims do, most of the worship happens in the morning aside from the obligatory 5 prayers, it is also a Sunnah to take a shower before going to the Eid prayer. There are other dhikr (remembering God with recitations) that can be recited after every salat for example.
On eid al Adha there is the slaughtering of an animal for consumption, of which a third is given to charity, before this the Muslim does not cut his nails or his hair (if I recall) during the ten days before the slaughter if I recall. There is also zakat al fitr which is payed on Eid al fitr which is an amount of money given in charity, the amount is determined by the local Islamic authorities (i.e. the masjid for example), usually it is ten dollars here. This is different than the 2.5% of salary zakat Muslims have to pay yearly.