What does "Eid Mubarak" mean?

The meaning of each word

• Eid: an Arabic word for a festival or celebration that comes back again and again. It comes from a root meaning "to return," because the joy returns every year.
• Mubarak: "blessed," from barakah, which means blessing, goodness that grows and lasts.

So "Eid Mubarak" is a short prayer as much as a greeting: may this celebration be full of blessing for you.

Why Muslims have Eid at all

When the Prophet ﷺ came to Medina, the people had two days of games left over from pre-Islamic times. He said, "Allah has substituted for them something better than them, the day of sacrifice and the day of the breaking of the fast" (Sunan Abi Dawud 1134). Eid al-Fitr follows Ramadan, and Eid al-Adha comes during the Hajj season. See What is the difference between Eid al-Fitr and Eid al-Adha?

Other Eid greetings you'll hear

Whatever greeting you hear, the Quran asks you to answer it well:

• Taqabbal Allahu minna wa minkum: "May Allah accept (our deeds) from us and from you." This is reported from the Companions, who said it to each other when they met on Eid. It isn't a hadith of the Prophet ﷺ, but scholars consider it a good, well-established greeting.
• Eid Sa'id: "Happy Eid" in Arabic.
• Kul 'am wa antum bikhair: "May you be well every year," common in Arab countries.
• Bayram Mubarak: used by Turkish and Balkan Muslims.
• Selamat Hari Raya: used in Malaysia, Indonesia and Singapore.

Any of these is fine. Use whatever your local community says, or plain English: "Happy Eid!" works too.

How and when to say it

• After the Eid prayer, people greet and often hug those next to them (men with men, women with women).
• By message: texting "Eid Mubarak" to friends and family on the morning of Eid is normal.
• To non-Muslims: yes, you can share it with them. See Can I say "Eid Mubarak" to non-Muslims?
• Pair it with salam: "Assalamu alaikum, Eid Mubarak!" covers both. See Can I say "Assalamu alaikum" if I'm not Muslim?