The short answer is, no, oats do not contain gluten. However, oats are at a high risk for cross-contamination since they are often grown near wheat or processed on machinery that also processes wheat. For that reason, you have to be careful with oats, especially if you have celiac disease or are highly sensitive to gluten.
To be on the safe side, it's best to only buy oats that are clearly marked "gluten free." Bob's Red Mill offers gluten free oat bran, rolled oats, steel cut oats, quick rolled oats, and oat flour.
The Celiac Foundation recommends that celiacs eat oats only in moderation-- and this is the clearly gluten-free, non-contaminated oats. Further, they recommend checking with your doctor before you eat oats.
So there you have it: while oats do not actually contain gluten, they are sometimes contaminated, so it's best to buy only the gluten free varieties. I hope that helped clear up the confusion.