How do the cardinals vote?

Answer 1

This is taken from the Catholic News Service:
Only cardinals under the age of 80 can vote in the conclave; older cardinals do not enter the Sistine Chapel. In theory, any baptized male Catholic can be elected pope, but current church law says he must become a bishop before taking office; since the 15th century, the electors always have chosen a fellow cardinal.

Each vote begins with the preparation and distribution of paper ballots by two masters of ceremonies, who are among a handful of noncardinals allowed into the chapel at the start of the session.

Then the names of nine voting cardinals are chosen at random: three to serve as "scrutineers," or voting judges; three to collect the votes of any sick cardinals who remain in their quarters at the Domus Sanctae Marthae; and three "revisers" who check the work of the scrutineers.

The paper ballot is rectangular. On the top half is printed the Latin phrase "Eligo in Summum Pontificem" ("I elect as the most high pontiff"), and the lower half is blank for the writing of the name of the person chosen.

After all the noncardinals have left the chapel, the cardinals fill out their ballots secretly, legibly and fold them twice. Meanwhile, any ballots from sick cardinals are collected and brought back to the chapel.

Each cardinal then walks to the altar, holding up his folded ballot so it can be seen, and says aloud: "I call as my witness Christ the Lord who will be my judge, that my vote is given to the one who before God I think should be elected." He places his ballot on a plate, or paten, and then slides it into a receptacle, traditionally a large chalice.

When all the ballots have been cast, the first scrutineer shakes the receptacle to mix them. He then transfers the ballots to a new urn, counting them to make sure they correspond to the number of electors.

The ballots are read out. Each of the three scrutineers examines each ballot one-by-one, with the last scrutineer calling out the name on the ballot, so all the cardinals can record the tally. The last scrutineer pierces each ballot with a needle through the word "Eligo" and places it on a thread, so they can be secured.

After the names have been read out, the votes are counted to see if someone has obtained a two-thirds majority needed for election -- or a simple majority if the rules are changed later in the conclave. The revisers then double-check the work of the scrutineers for possible mistakes.

Answer 2

I know they begin off by praying to the Holy Spirit for Guidance in their choice for a new Pope. Most likely they do it behind a ballot box, like a regular election.