Understanding your users is critical to gamification design. While you need to undergo a design research process to understand each context you are gamifying, it is useful to categorise different ways people play.
Probably the most well known categorisation is Bartle's taxonomy. It classifies four types of players, based on their character and behaviours during gameplay. The Bartle taxonomy of player types was originally used for video gamers and has since been extended to other types of play. It's based on character theory and focuses on how players interact and behave in a game environment. While all players exhibit all characteristics, most have a dominant type.
Killers thrive on competition and enjoy competing and fighting with other players. Killers enjoy pitting their skill against another, either individually or as a team.
Achievers value gaining points, levels, equipment and other tangible measurements of success. Achievers tend to like collecting these accolades for the prestige, not necessarily for gameplay benefit.
Socialisers gain most enjoyment by interacting and collaborating with others. Socialisers enjoy forming and building relationships.
Explorers have a preference to discover, whether it be new areas, creating maps or hidden places and features, eastern eggs and even glitches. Explorers like playing at their own pace.
There is some debate about the generalisability of these four types, and they do not cover the distinct needs of users in a particular context, but they do offer a good heuristic to ensure your gamification design is meeting the needs of different players.