Of all the basic and traditional Catholic prayers, I only know four from memory, assuming you count the sign of the cross as its own prayer.
SIGN OF THE CROSS
In the Name of the Father, and of the Son,
and of the Holy Spirit, Amen.
This is typically said at the beginning and end of prayers, and is usually accompanied by moving one's hand in a certain motion representing a cross. Its origins date back to the second or third centuries. More information here and here.
OUR FATHER/ LORD'S PRAYER
Our Father, Who art in Heaven, hallowed be
Thy Name. Thy Kingdom come. Thy Will be done,
on earth, as it is in Heaven. Give us this day our
daily bread and forgive us our trespasses as we
forgive those who trespass against us; and lead us
not into temptation, but deliver us from evil. Amen.
This one everybody knows, though we do not have the "For thine is the kingdom, and the power, and the glory, for ever and ever," at the end. It is added in the Mass as an independent element. What I did not know until just now is that it used to be known only as the Pater Noster because it was always said in Latin. I guess that makes sense.
More information here.
Hail Mary, full of grace, the Lord is with thee.
Blessed art thou among women, and blessed is
the fruit of thy womb, Jesus.
Holy Mary, Mother of God, pray for us sinners,
now, and at the hour of our death. Amen.
Now we get to one that sometimes gives Catholics a bad rap, thinking that we are worshiping Mary. If we read the prayer carefully, we can see that we are "praying" in its original intent - we are asking. We are asking that Mary pray for us.
The first three lines are clearly from the Bible itself, with only "Jesus" added. We do believe that she is the Mother of God, and we further believe that people who have died and are in Heaven continue to care about us and want to pray for us so we ask her to pray for us too. Just like I might ask one of you all to pray for me. The difference being, of course, she's Jesus' mom. I will refrain from discussion on any of the other dogmas regarding Mary, but that's the prayer.
More information here and here.
This is also set to beautiful music - Ave Maria. (And I will admit to not actually connecting the prayer to the song until at least two years ago because I am a doof.)
Glory be to the Father, and to the Son, and to the Holy Spirit, as it was in the beginning, is now, and ever shall be, world without end. Amen.
Short and to the point! More information here and here.