The 1st of Shawwal celebrates completing Ramadan. Before its prayer, every Muslim gives Zakat al-Fitr so the poor celebrate too. The day begins with ghusl, best clothes, eating dates, and the takbirat.
The 10th of Dhul-Hijjah, during Hajj days, commemorates Ibrahim's readiness to sacrifice his son in obedience to Allah. Those able offer the udhiyah (sacrifice) and share the meat with family, neighbors and the poor.
The Eid prayer is held in congregation with a khutbah; families visit, children play, gifts are given — celebration in Islam is gratitude to Allah, joy without disobedience.