The reputation system is a way to show the helpfulness, or humor of a person. For a Republic member, if they made you laugh or helped you out, you would press the + Reputation symbol. For an Empire member that did the same, you would press the - Reputation button.
More + Rep for a Republic member the better.
More - Rep for an Empire member the better.
These have no true influence over your standing in the community, just a symbol of appreciation from your fellow members.