In its most basic form, almsgiving is simply charity, giving something that we have to someone else. As the Holy Father notes, it is has two objects: it is "a specific way to assist those in need and, at the same time, an exercise in self-denial to free us from attachment to worldly goods."
As Christians, "we are not owners but rather administrators of the goods we possess," and we are bound to use them for the good of others as well as for our own good. Almsgiving is true love of neighbor: "it is a concrete expression of charity, a theological virtue that demands interior conversion to love of God and neighbor, in imitation of Jesus Christ, who, dying on the cross, gave His entire self for us."
We've moved, then, from simply writing a check to a charity to something much more important: the imitation of Christ, which consists "not so much in giving a part of what we possess, but our very selves." Thus, the Holy Father concludes, the ultimate goal of almsgiving is not simply to meet the material needs of our neighbors, but to introduce them to Christ through our actions: "In giving alms, we offer something material, a sign of the greater gift that we can impart to others through the announcement and witness of Christ, in whose name is found true life."
You can read the entire text of Pope Benedict's message for Lent 2008 at the Vatican's website.