Expertise (prior to Season 6: Bridge Officer Skill Points) is used as a cost for applying a Training Manual on a bridge officers, teaching them new abilities. Cost of Expertise 
is increased with the number of abilities the BOff already possesses. Expertise is also needed as a contribution to certain projects of the Reputation System, as well as the Fleet Advancement System.
Just like Experience Points 
, expertise is earned indefinitely by playing almost any content of Star Trek Online, e.g., defeating NPC or player enemies, completing missions/assignments, and so on.