Is peat a type of fossil fuel?

No, it is not a fossil. It is however the first stage of producing a fossil fuel. Peat is thick vegetation that has died and built up in layers in a wet environment which doesn't decay normally (aerobic decay) because of a lack of oxygen in the layers. Anaerobic decay, or a very slow process of composting, occurs in a peat bog which eventually becomes a coal deposit, over hundreds of thousands or millions of years. Instead of creating carbon dioxide, like aerobic bacteria, and fungi produce in normal decay, Anaerobic bacteria produce methane gas as they break down the peat. This produces lignite or brown coal, which as it ages and the lignite compacts eventually, giving up moisture and nitrogen, become progressively harder and darker coals.