Unlike Star Trek - Armada 1 and 2, Armada 3 is not an independent game from Activision, but a modification for the game Sins of a Solar Empire: Rebellion, released by Stardock. The mod had its first stable release on Feb 27, 2014.
Sins of a Solar Empire: Rebellion must already be installed via Steam. Start the game via Steam. The Sins launcher starts. Once shown, select the game on the left hand side and click on the Play button. In the games main menu, select Options, then Mods. Us the Show Mod Path Button. It will show you a path very similar to this one:
C:\Users\<user>\Documents\My Games\Ironclad Games\Sins of a Solar Empire Rebellion\Mods-Rebellion v1.85. Note it, or
TAB outside of the game, to open the folder right away. Inside this folder, the mods will have to be placed. The mod itself can be downloaded from https://www.moddb.com/mods/star-trek-armada-3/downloads?sort=dateup-desc. You will recieve a zip archive, containing the mod data. You will find a folder
STA3_UPRISING inside it. Extract it into the above mentioned folder. Once done, (re)start the game, get back to the Mods menu and activate the newly found mod.
This video also describes the procedure.
The game is strictly a multiplayer mod, meaning there is no campaign of any kind. Like the original game, the mod also has the ability to save a multiplayer match and continue it at another time. It is however important, that all players use the same version during continuration as they did during the saving of a match.