Reputation

There are a myriad of ways to gain (and lose!) reputation with the various factions you'll meet. The easiest way to increase your rep is simply to travel to different locations. Upon entering the faction’s system, your reputation will increase, albeit marginally. Boosting faction ships or participating in trade will net larger reputation gains. Completing trade missions will have an even greater affect on your reputation.

Draining a ship's engines will lower your standing with that faction. However, if that faction has an enemy, your actions may end up increasing your reputation with that faction.

Gemini Station
Your reputation with Gemini Station can be increased by traveling & selling goods to the station, boosting Gemini Station ships, and by attacking or draining pirate ships.

Earth
Your reputation with Earth can be increased by traveling and selling goods to Earth, boosting Earth ships, and by attacking or draining Martian ships.

Mars
Your reputation with Mars can be increased by traveling and selling goods to Mars, boosting Martian ships, and by attacking or draining Terran ships.

Jupiter
Your reputation with Jupiter can be increased by traveling, selling, and buying goods from Jupiter.

Eris
Your reputation with Eris can be increased by traveling, selling to and buying goods from Eris.

Mercury
Your reputation with Mercury can be increased by traveling to Mercury Station and buying & selling goods.