Another area of controversy. (Although it's good to be posting about these areas for those who are fairly new to all this.)
Oats do not contain the same type of gluten protein that celiacs react to. The structure of the oat protein molecules is similar to that of whet, rye, and barley, but different in a few respects.
The problem over the years has come from contamination of oats with those other proteins. Often, oats are grown in fields near (or even within) fields in which these gluten-containing grains are grown, and it is a simple matter for there to cross-contamination by wind, by pollinators, but perhaps most importantly, during the processing of the grain into food.
The key, I believe, is to consume oats harvested/milled from a facility of machine line DEDICATED to oats--not one that is also used to process on of the other grains. Any small amount of contamination risks a reaction in the sensitive. It used to be very hard to find oats processed in such a manner, but it has gotten easier with the interest in gluten-free eating, and oat products will usually say whether they have been created using a "dedicated" oat processing facility. (If the product doesn't indicate that, be suspicious.)