Facebook is the world's largest social networking database. At the moment Facebook has millions of users spread across the world, and this includes people from extremely varied demographics. Considering the length and breadth of Facebook's user database, it seems almost criminal not to consider Facebook as a major marketing tool!
Facebook groups are a great way to advertise your brand, be it a place, service, product or concept. Groups have the potential to grow rapidly as members can invite their friends to join. There are various privacy settings that can control how a group works. 'Pages' on the other hand, are open to everyone and are also a good way to attract members.
Any niche database of users is always a target for advertising and marketing. For example, a Facebook group entitled Guide to singles sites and dating is obviously a potential target for dating applications, or matrimonial and dating websites!
Facebook pages are a more viable option for big brands, companies or artists. For example, a music band is better off with a fan page than a small intimate group. But a group of nature lovers looking to protect the local bird habitat near their town may do well in forming a group.
Facebook groups can be better understood when we think of them as fellowships or clubs of sorts. They are based around a common purpose and usually have some system of administration and membership.