The Eid Prayer is wajib (obligatory) according to the majority of scholars. The Prophet (peace be upon him) performed it consistently and commanded all the people to go out for it.

He (peace be upon him) even specifically commanded all of the women to attend; Umm `Atiyyah (may Allah be please with her) narrated that the Prophet (peace be upon him) commanded:

Let the free women, the virgins, and the menstruating women go out to attend the Eid prayer, and witness the good and the supplications of the believers. As for the menstruating women, they should stay away from the musalla. (i.e., should not participate in the prayer itself).

Sheikhul-Islam Ibn Taymiyyah said:

"... Thus we lean towards the opinion that Salatul-Eid is fard `Ayn (individual obligation) on each individual. This is the opinion of Abu Hanifah and others, and is one of the two reported opinions of ash-Shafi`i and of the school of Ahmad.
The opinion held by some people, that it is not wajib, it is extremely unlikely, because it is one of the greatest rites of Islam; more people gather for it than for Friday; and the Takbir has been legislated for its sake. The opinion of those who say that it is fard kifayah (communal obligation) is unjustified..."

Ash-Shawkani said:

"Note that the Prophet (peace be upon him) consistently performed this prayer on every Eid, never neglecting it. He commanded the people to go out for it; he even commanded the free women, the virgins, and the menstruating women to go out, instructing the latter to refrain from praying, in order for all of them to witness this good and the Muslims' supplications. He (peace be upon him) further commanded the woman who did not own a jilbab (outer overall garment for women) to borrow her friend's.
All of this proves that this prayer is wajib on every individual, not only a kifayah obligation. Commanding the people to go out for it entails the command to perform the prayer (for those who have no excuse). This is the content of his (peace be upon him) address, because going out is a means for performing the prayer (an end); if the means is obligatory, then the end is. And this obligation applies to men even more so that women.
Further evidence is that it overrides the Friday prayer when both occur on the same day. That which is not wajib cannot override that which is.
Furthermore, it is confirmed that the Prophet (peace be upon him) performed it consistently in Jama`ah (congregation) from the time that it was legislated until he died. To this is added his command to people to go out for it."

Commenting on the Hadith of Umm `Atiyyah, our sheikh al-Albani states:

"The Prophet's (peace be upon him) command mentioned here indicates obligation; when it is wajib to go out for it, it is wajib to pray it, because of its higher priority. Thus the truth is that it is an obligation - not a mere recommendation."