In its natural form, both peanuts and peanut butter are gluten-free.
Many store-bought brands of peanut butter are also gluten-free, with gluten-containing peanut butter tending to be the exception rather than the rule. But beyond the basics of peanuts, oil, sweetener and salt, there can be additional ingredients, particularly in more-processed brands. Rarely, these added ingredients can be gluten-containing, so always be on the lookout for the gluten-free label.
Additionally, some brands may be processed in facilities that also process wheat. This information will almost always be on the jar, so please consider your risks of cross-contact before buying. However, when safe manufacturing protocols are in place, it is possible to safely manufacture gluten-free food for people with celiac disease in a shared facility.
At most grocery stores there are many varieties of safe, gluten-free peanut butter, so if you can’t determine whether a product is gluten-free or not, it’s best to stick to one that is labeled as such.