Praying for the dead is an act of charity. This is not only a pious practice, but is indeed a duty for us all. The more you assist the holy souls with your alms, the more you help yourself. We especially have the duty to pray for our loved ones and relatives.
In addition to having a Mass said on the anniversary of a loved one's death, there exists an old Christian custom of fasting, burning a candle for 24 hours, and praying for them on this day (this custom was adopted by Jews in the Middle Ages; they call it "Jahrzeit" as did the medieval German Christians.) Any orthodox traditional or improvised prayer can be prayed at this time, but the Rosary is always appropriate and can be prayed now (or any time) for the dead, offered in the same way it is at Catholic funerals -- i.e., with the Fatima Prayer replaced (or in addition to) the Eternal Rest Prayer. The Eternal Rest prayer is also prayed by itself, offered any time for the dead, when visiting grave sites, and when passing cemeteries. It is a prayer every Catholic should know by heart:
Eternal rest grant unto them, O Lord; and let perpetual light shine upon them. May they rest in peace. Amen.
Réquiem ætérnam dona ei (eis) Dómine; et lux perpétua lúceat ei (eis). Requiéscat (Requiéscant) in pace. Amen.
We may also pray for our dead every time we eat if we pray the After Meal Blessing, another prayer every Catholic should know by heart:
We give Thee thanks for all Thy benefits, O Almighty God, Who livest and reignest forever. And may the souls of the faithful departed, through the mercy of God, rest in peace. Amen.
Agimus tibi gratias, omnipotens Deus, pro universis beneficiis tuis, qui vivis et regnas in saecula saeculorum. Fidelium animae, per misericordiam Dei, requiescant in pace. Amen.
The Penitential Psalms -- i.e., Psalms 6, 31, 37, 50, 101, 129, 142 -- are also prayed for the dead, especially the 129th Psalm, known as De Profundis.