For all months except August (when the Vatican offices are closed), tickets are required for Papal Audiences. Tickets are always free.
If you need less than 10 tickets then you can simply ask the Swiss Guards for some. They have a stash of tickets explicitly for pilgrims who ask for them without a reservation. The Swiss Guard stationed at the Bronze Door (just to the left of the metal detectors of St Peters and then first right) have some to give out, as do the Swiss Guard at Porta Anna.
However, please note that the Swiss Guard will only have tickets for the Papal Audience or Papal Mass the day before the event. Tickets will be given out from 3pm until 7pm the day before the scheduled Audience, and from 7am on the day.
For groups greater than 10, or if you wish to reserve your tickets in advance, you have 3 options.
TICKETS FOR THE PAPAL AUDIENCE ARE ALWAYS FREE.