History and specifications
In the 23rd century, USS Enterprise personnel had body armor that zipped itself up, leaving no seam. This armor was like silk, but an impact would render it rigid. (TOS novel: Planet of Judgment, Heritage Models RPG module: Adventure Gaming in the Final Frontier)
On 03 April 2385, Department of Temporal Investigations agent Meyo Ranjea attempted to apprehend the criminal Daiyar on the spaceport of the Qhembembem Outpost. When his phaser fire left her unfazed, Ranjea deduced she wore body armor under her cloak. (DTI eBook: Shield of the Gods)
By the year 2409, the Federation, Klingon Empire and Romulan Republic offered body armor as standard equipment to their starship commanders and their bridge officers. The body armor was invisible by default, integrated with the uniform or other clothing, just like apersonal shield and kits. Tailors of the Khitomer Alliance were able to add visible body armor components to an officer's outfit.
When the USS Miranda-B responded to a Borg invasion of Vega IX, Ensign Byarnez was authorized by Commander Kelly to equip the Miranda away team defending Vega Colony with a phaser sniper rifle, personal kit, and body armor. (STO - Assimilation mission: "Gather Equipment")
The Khitomer Alliance members differentiated between basic types of body armor: energy armor, physical armor, and utility armor. Other categorizations included energy dampening, energy harnessing, physical augmentation, recoil compensating, and polyalloy weave armor. (STO - Klingon War missions: "Tutorial", "Welcome to Earth Spacedock")
25th century environmental suits also included armor-like protection from physical and energy damage. Environmental suits deployed by the Alliance on Nukara Prime protected combatants in the conflict with unarmored Tholians. (STO - Tholian Incursion mission: "Hot Lead")