Troops and characters usually participate in challenges, but do not interact with one another directly. Instead, characters interact with characters and troops interact with troops. Fleets represent the primary military capability of any race. An individual faction can only have one copy of any give fleet out at a time. Some fleets can only be played if your faction is from a particular race.
Groups represent large numbers of minor characters or some organization.
Some group cards can only be played if you are playing a specific race. Most group cards don't have abilities like characters, but have effects on the gameplay when you use them. Advantages are long term effects that a character, the player, or troops can gain. Until something has you discard it, the advantage keeps on churning out its effect. If the card an advantage is attached to leaves play, so does the advantage.