I guess most of the conversations revolves around money.
A soldier usually needs permission from his captain before entering into any business venture. He always keeps his captain updated about his business and wherabouts. Even if the soldier wants to go away on vaccation a couple of days, he have to ask permission from his captain. The captain needs to know where to reach out for him if he´s needed for anything.

The consigliere settles beefs and disputes. So he listens to the parties both sides and make out a ruling.
The beefs usually concerns money, cheating, loans, settlements of business interests and such things.
But a beef could also be about some abuse or failing of showing proper respect.

If you are interested in all this, I would recomend (and I think Mukremin also recomended) Jerry Capeci´s book "The Complete Idiot´s Guide To The Mafia". It´s an interesting read.