Well, let's first discuss the precepts involved here:
CCC 2042 The first precept ("You shall attend Mass on Sundays and holy days of obligation and rest from servile labor") requires the faithful to sanctify the day commemorating the Resurrection of the Lord as well as the principal liturgical feasts honoring the Mysteries of the Lord, the Blessed Virgin Mary, and the saints; in the first place, by participating in the Eucharistic celebration, in which the Christian community is gathered, and by resting from those works and activities which could impede such a sanctification of these days.
The second precept ("You shall confess your sins at least once a year") ensures preparation for the Eucharist by the reception of the sacrament of reconciliation, which continues Baptism's work of conversion and forgiveness.
The third precept ("You shall receive the sacrament of the Eucharist at least during the Easter season") guarantees as a minimum the reception of the Lord's Body and Blood in connection with the Paschal feasts, the origin and center of the Christian liturgy.
Can. 920 §1. After being initiated into the Most Holy Eucharist, each of the faithful is obliged to receive holy communion at least once a year.
§2. This precept must be fulfilled during the Easter season unless it is fulfilled for a just cause at another time during the year.
Code of Canon Law 920What Easter Duty is NOT
So, first off we must understand that "Easter Duty" is NOT merely the requirement to go to Mass once a year at Eastertide. This is a misnomer held by many a Catholic who doesn't really know their faith. Many "Catholics" show up for Mass once per year on Easter Sunday (some go twice per year with Easter and Christmas), my challenge to those was posted in 2006, (click here). Easter Duty does not imply that Catholics are only required to attend Mass once per year! Those who think this way are ignoring the First Precept, wherein faithful Catholics are required to participate in Mass on ALL SUNDAYS and Holy Days of Obligation. To only go to Mass once per year grossly ignores the First Precept.
What Easter Duty IS
Easter Duty refers to the Third Precept, mentioned above. It is the requirement to receive the Eucharist at least once during the Easter Season. In years past reception of the Eucharist was not an every Sunday affair and especially not daily reception - as many modern Catholics are accustomed to. Reception of Eucharist could be once per quarter, once per 6 months or even once per year. The Church decreed that faithful Catholics must receive Eucharist at least once during the season of Easter, or Eastertide.
What is Eastertide?
The season of Easter, or Eastertide, begins on Easter Sunday and lasts through Pentecost Sunday. There's no rush to get ones Easter Duty done on Easter Sunday. You have from the First Mass of Easter, during the Easter Vigil, through the last Mass of Pentecost Sunday.
To summarize - faithful Catholics are required to attend and participate in Mass on all Sundays and holy days of obligation (the First Precept). At least once per year, during Eastertide, faithful Catholics are to receive the Eucharist (the Third Precept). In order to licitly receive the Eucharist, faithful Catholics must be spiritually prepared, that is, they cannot have mortal sin on their soul (to not fulfill the First Precept not attending Mass every Sunday is a mortal sin), thus the necessity to go to Confession prior to Eucharist arises, and we are required to confess our sins at least once per year too (the Second Precept). So the "Easter Duty" or "Easter Obligation" is to receive Eucharist at least once between the First Mass of Easter (during Easter Vigil) and the final Mass on Pentecost Sunday.