Eid is the name for the two Muslim holidays; Eid ul-Adha and Eid ul-Fitr. "Eid Mubarak" is a common phrase among Muslims; it means "(Have a) Blessed Eid" You don't have to be Muslim to wish someone "Eid Mubarak," and is a great way to meet new people during the holidays.
Often this is accompanied by a handshake, or more often a special Eid hug; where you hug the person and lean to the right, then hug again while leaning to the left side and then the right side again, all while not really letting go.