Outside of the Mass, prayers of petition are the type of prayer with which we are most familiar. In them, we ask God for things we need—primarily spiritual needs, but physical ones as well. Our prayers of petition should always include a statement of our willingness to accept God's Will, whether He directly answers our prayer or not. The Our Father is a good example of a prayer of petition, and the line "Thy will be done" shows that, in the end, we acknowledge that God's plans for us are more important than what we desire.
Prayers of expiation, in which we express sorrow for our sins, are one form of prayers of petition. In fact, the first form because before we ask for anything, we should acknowledge our sinfulness and ask God for His forgiveness and mercy. The Confiteor or Penitential Rite at the beginning of Mass, and the Agnus Dei (or Lamb of God) before Communion are prayers of expiation, as is the Act of Contrition.
In the silence of the heart God speaks. If you face God in prayer and silence,
God will speak to you. Mother Teresa (1910-1997)
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
For the kingdom,
the power and the glory are yours
now and for ever.
Act of Contrition
O my God,
I am heartily sorry for having offended Thee, and I detest all my sins,
because I dread the loss of Heaven
and the pains of hell,
but most of all because they offend Thee,
Who are all good and deserving of all my love. I firmly resolve
with the help of Thy grace
to confess my sins,
to do penance,
and to amend my life.
Prayer of Adoration and Confession
My Dear God,
I am sorry when I get so carried away with the busyness of each day,
never stopping to dwell in your love.
I am sorry when I allow the stresses and demands of work or family to erode and wear away my hope.
I am sorry when I'm caught up in conflict and frustration,
when I harbour resentment, instead of giving out forgiveness.
I'm sorry when I seek to be fulfilled by food, drink or entertainment,
Yet fail to feast on your truth.
I'm sorry when I desire the wrong things, blinded by the media, and forget to engage with the real treasures of love and justice.
You are all love, all hope, all forgiveness, all truth and all I really need.