Throughout the liturgical year, the Catholic Church makes certain changes to the Mass to reflect the liturgical season. Next to the change in the color of the priest's vestments, the absence of the Alleluia during Lent is probably the most obvious.
The Meaning of the Alleluia
The Alleluia comes to us from Hebrew, and it means "praise Yahweh." Traditionally, it has been seen as the chief term of praise of the choirs of angels, as they worship around the throne of God in Heaven. It is, therefore, a term of great joy, and our use of the Alleluia during Mass is a way of participating in the angels' worship. It is also a reminder that the Kingdom of Heaven is already established on earth, in the form of the Church, and that our participation in Mass is a participation in Heaven.
Our Lenten Exile
During Lent, however, our focus is on the Kingdom coming, not on the Kingdom having come. The readings in the Masses for Lent and in the Liturgy of the Hours focus heavily on the spiritual journey of Old Testament Israel toward the coming of Christ, and the salvation of mankind in His death and resurrection.
We, too, are on a spiritual journey, toward the Second Coming and our future life in Heaven. In order to emphasize that journey, the Church, during Lent, removes the Alleluia from the Mass. We no longer sing with the choirs of angels; instead, we acknowledge our sins and practice repentance so that one day we may again have the privilege of worshiping God as the angels do.
The Return of the Alleluia at Easter
That day come triumphally on Easter Sunday—or, rather, at the Easter Vigil, on Holy Saturday night, when the priest chants a triple Alleluia before he reads the Gospel, and everyone present responds with a triple Alleluia. The Lord is risen; the Kingdom has come; our joy is complete; and, in concert with the angels and saints, we greet the risen Lord with shouts of "Alleluia!"