As St Therese says, “...as long as you recite the Rosary, God will not abandon the world, because it is powerful on His Heart”. 

Italian organisation CENTRO CULTURALE LEPANTO has launched 1 million Rosary crusade utnil the end of conclave. Please sign up for the prayers HERE. Let us pray this Lent for the new Pope.
From Papal Encyclicals online, the Pope Paul VI Encyclical 'Election of the Roman Pontiff' to read HERE , and from Vatican website, Encyclical of the Pope John Paul II 'Universi Dominici Gregis' HERE and the motu proprio of Pope Benedict 16 'Normas Nonnullas' HERE
The list of Cardinal Elelctors HERE
News from Vatican HERE
The first meeting of General Congregation is scheduled on Monday the 4th of February which means the Cardinals may vote for the starting day of Conclave on Monday see here

Latest news, conclave is scheduled on Tuesday, March 12th in the afternoon according to Vatican News
 

Cardinal Electors group photo (after Rinnovamento Nella Fede Cattolica), and one of them may be elected for the next Pope.

Voting explained:

Smoke times: 12th of March, only one ballot and results around 7-7:30pm (Rome local time). Since 13th of March, four rounds of voting. We may see first smoke in the morning at around 11:30-noon and in the evening, between 6pm  and 7pm.

The smoke may be watched directly with Vatican player
Some of the history of Papal conclaves HERE

13th of March 2013 - 18.05 London time - HABEMUS PAPAM!

The new Pope is Cardinal Jorge Bergoglio, Jesuit from Argentina, Pope Francis I, the name taken possibly after great Jesuit Saint, St Francis Xavier.
Update: I was wrong, the new pope took the name Francis in honor of St. Francis of Assisi because he is a lover of the poor, according to Vatican spokesman the Rev. Tom Rosica said. Also, the new pope should be known as Pope Francis, not Pope Francis I.

First Urbi et Orbi

I hope and pray the new Pope, Pope Francis will be pleasing to God. I have devotion to St Ignatius, the founder of Jesuits, as he is my patron Saint. AMDG 

Prayer for the Sovereign Pontiff V. 

Oremus pro Pontifice nostro Franciscum. 

R. Dominus conservet eum, et vivificet eum, et beatum faciat eum in terra, et non tradat eum in animam inimicorum eius. (Ps 40:3) 

Pater Noster, Ave Maria,

Deus, omnium fidelium pastor et rector, famulum tuum Franciscus, quem pastorem Ecclesiæ tuæ præesse voluisti, propitius respice: da ei, quæsumus, verbo et exemplo, quibus præest, proficere: ut ad vitam, una cum grege sibi credito, perveniat sempiternam. 

Per Christum, Dominum nostrum. Amen. V. 

Let us pray for Francis, our Pope. 

R. May the Lord preserve him, and give him life, and make him blessed upon the earth, and deliver him not up to the will of his enemies. (Ps 40:3) 

Our Father, Hail Mary 

O God, Shepherd and Ruler of all Thy faithful people, look mercifully upon Thy servant Francis, whom Thou hast chosen as shepherd to preside over Thy Church. Grant him, we beseech Thee, that by his word and example, he may edify those over whom he hath charge, so that together with the flock committed to him, may he attain everlasting life. Through Christ our Lord. Amen.

"Yesterday, it would seem, we received a Pope who is truly a servants of the servants of God as the world best knows it: a man who is not afraid to approach the weak, the sick and the poor, who shuns pomposity and vanity and, as we soon learned, chose to take the bus with the other cardinals back to the Domus Sanctae Marthae, instead of taking the limousine that was waiting.

But that humility should not be taken for weakness or even simplicity. As his chosen papal name indicates, underneath the simplicity of his appearance and actions, not unlike his two immediate predecessors, lies a person of great strength and faith. Whereas Benedict XVI was the professor who taught us about the faith, Francis will be the older brother who walks with us and shows us the way in love and charity." comment on Papal election from In Coelo et in Terra blog