A badge was an insignia which could be a symbol of authority, responsibility or membership of an organization. (DS9: "Trials and Tribble-ations")
The badge worn by many generations of Starfleet personnel was shaped like an arrowhead. Like the uniform, it showed the division in which one was serving. In the 24th century, this badge was integrated with the communicator to become the combadge.
In the novel Federation, written by Judith and Garfield Reeves-Stevens, the shape was explained as indicative of the warp threshold boundary. The star in the middle of the badge represented the speed of light, the upper edge of the badge was a graph of normal energy versus speed (indicating that it was impossible to reach the speed of light without infinite energy) and the lower edge the graph with warp engines taken into account.