Good question, but hard to answer. Average days off vary dramatically by company and seniority within that company. At my company, most people probably average around 14 days off per month. On the low side we have people with 10-11 days off, and on the high side people with 18-19 days off.
We are paid based on "Block" time. This is essentially the time from when the parking brake is released to when it is reset at your destination. While you are preflighting, getting clearances, checking weather, talking to passengers, or sitting around the airport for 3 hours at a time you are not getting paid. This is why our hourly rates sound pretty spectacular to someone in a non-aviation job. "Wow, you make $60/hour?!" In a regular job, that would be around $120,000/year. In the aviation world, it's around $60,000. Keep in mind, you probably won't ever hit that pay scale until you are a captain at a regional airline.
Keep in mind that while 14 days off sounds like a lot ("The average joe only gets 10 days off a month in a 9-5 job, right?") it is very different in the airline lifestyle. While your counterpart in the corporate world goes home to his/her family at 5:00pm, you will generally spend your 3-4 day work week gone the entire time. When you get home, you will probably spend a good portion of at least one of your days off running around doing all the errands most people do before or after work; laundry, paying bills, mowing the lawn, talking to your spouse and children, etc.
For me to have the same quality of life I've had with non-aviation jobs, I need a minimum of about 15 days off in my airline job.
p.s. Mojo's information regarding "timing out" at the end of the year is correct at most airlines, but not all. SkyWest, for example, will not pay you guarantee for the remainder of the year if you time out early. Time off with no pay in return for working your heinie off. Merry Christmas!!