Some guidance on what to expect on March 12, from Fr. Thomas Reese:
On the morning the conclave begins, the cardinal electors celebrate Mass in St. Peter’s Basilica. In the afternoon they gather in the Pauline Chapel in the Apostolic Palace and solemnly process to the Sistine Chapel. The cardinals take an oath to observe the rules laid down in the papal constitutionUniversi Dominici Gregis, especially those enjoining secrecy. They also swear not to support interference in the election by any secular authorities or “any group of people or individuals who might wish to intervene in the election of the Roman pontiff.” Finally, the electors swear that whoever is elected will carry out the “munus Petrinum of pastor of the universal church” and will “affirm and defend strenuously the spiritual and temporal rights and liberty of the Holy See.”
After the oath is taken, everyone not connected with the conclave is ordered out with the Latin words “Extra omnes,” “Everybody out!” The Sistine Chapel and the Domus Sanctae Marthae are then closed to unauthorized persons by the camerlengo, Cardinal Tarcisio Bertone. Outside the conclave, the camerlengo is assisted by the sostituto (Archbishop Giovanni Becciu) of the Secretariat of State, who directs Vatican personnel to protect the integrity and security of the conclave.
After everyone else leaves, an ecclesiastic chosen earlier by the college of cardinals gives a meditation “concerning the grave duty incumbent on them and thus on the need to act with right intention for the good of the universal church, solum Deum prae oculis habentes [having only God before your eyes].” When he finishes, he leaves the Sistine Chapel with the master of papal liturgical ceremony so that only the cardinal electors remain. The time in the chapel is for prayer and voting in silence, not campaign speeches. Negotiations and arguments are to take place outside the chapel. If they wish, the cardinals can immediately begin the election process and hold one ballot on the afternoon of the first day. If no one receives the required two-thirds vote in the balloting on the afternoon of the first day, we get black smoke and the cardinals meet again the next morning.
UPDATE: CNS has a superlative video explanation below.